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Chapter 1 COMPUTATIONAL FLUID DYNAMICS 1.1 Introduction [2] Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with surfaces defined by boundary conditions. With high-speed supercomputers, better solutions can be achieved. Ongoing research, however, yields software that improves the accuracy and speed of complex simulation scenarios such as transonic or turbulent flows. Initial validation of such software is performed using a wind tunnel with the final validation coming in flight tests. 1.1.1 Background and history Fig. 1.1 A computer simulation of high velocity air flow around the Space Shuttle during re entry. 1
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Chapter 1

COMPUTATIONAL FLUID DYNAMICS

1.1 Introduction[2]

Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses

numerical methods and algorithms to solve and analyze problems that involve fluid flows.

Computers are used to perform the calculations required to simulate the interaction of

liquids and gases with surfaces defined by boundary conditions. With high-speed

supercomputers, better solutions can be achieved. Ongoing research, however, yields

software that improves the accuracy and speed of complex simulation scenarios such as

transonic or turbulent flows. Initial validation of such software is performed using a wind

tunnel with the final validation coming in flight tests.

1.1.1 Background and history

Fig. 1.1 A computer simulation of high velocity air flow around the Space Shuttle during re entry.

Fig 1.2 A simulation of the Hyper-X scramjet vehicle in operation at Mach-7

The fundamental basis of almost all CFD problems are the Navier–Stokes

equations, which define any single-phase fluid flow. These equations can be simplified by

removing terms describing viscosity to yield the Euler equations. Further simplification,

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by removing terms describing vorticity yields the full potential equations. Finally, these

equations can be linearized to yield the linearized potential equations.

Historically, methods were first developed to solve the Linearized Potential

equations. Two-dimensional methods, using conformal transformations of the flow about

a cylinder to the flow about an airfoil were developed in the 1930s. The computer power

available paced development of three dimensional methods. The first paper on a practical

three-dimensional method to solve the linearized potential equations was published by

John Hess and A.M.O. Smith of Douglas Aircraft in 1967.

1.2 Basic equation used for determining the fluid flow[1]

1.2.1 Navier–Stokes equations

In physics the Navier–Stokes equations, named after Claude-Louis Navier and

George Gabriel Stokes, describe the motion of fluid substances. These equations arise

from applying Newton's second law to fluid motion, together with the assumption that the

fluid stress is the sum of a diffusing viscous term (proportional to the gradient of

velocity), plus a pressure term.

The equations are useful because they describe the physics of many things of

academic and economic interest. They may be used to model the weather, ocean currents,

water flow in a pipe and air flow around a wing. The Navier–Stokes equations in their full

and simplified forms help with the design of aircraft and cars, the study of blood flow, the

design of power stations, the analysis of pollution, and many other things. Coupled with

Maxwell's equations they can be used to model and study magneto hydrodynamics.

The Navier–Stokes equations dictate not position but rather velocity. A solution of

the Navier–Stokes equations is called a velocity field or flow field, which is a description

of the velocity of the fluid at a given point in space and time. Once the velocity field is

solved for, other quantities of interest (such as flow rate or drag force) may be found. This

is different from what one normally sees in classical mechanics, where solutions are

typically trajectories of position of a particle or deflection of a continuum. Studying

velocity instead of position makes more sense for a fluid; however for visualization

purposes one can compute various trajectories

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1.2.1.1 Properties

A. Nonlinearity

The Navier–Stokes equations are nonlinear partial differential equations in almost

every real situation. In some cases, such as one-dimensional flow and Stokes flow (or

creeping flow), the equations can be simplified to linear equations. The nonlinearity

makes most problems difficult or impossible to solve and is the main contributor to the

turbulence that the equations model.

The nonlinearity is due to convective acceleration, which is an acceleration

associated with the change in velocity over position. Hence, any convective flow, whether

turbulent or not, will involve nonlinearity. An example of convective but laminar

(nonturbulent) flow would be the passage of a viscous fluid (for example, oil) through a

small converging nozzle. Such flows, whether exactly solvable or not, can often be

thoroughly studied and understood.

B. Turbulence

Turbulence is the time dependent chaotic behavior seen in many fluid flows. It is

generally believed that it is due to the inertia of the fluid as a whole: the culmination of

time dependent and convective acceleration; hence flows where inertial effects are small

tend to be laminar (the Reynolds number quantifies how much the flow is affected by

inertia). It is believed, though not known with certainty, that the Navier–Stokes equations

describe turbulence properly

C. Applicability

Together with supplemental equations (for example, conservation of mass) and

well formulated boundary conditions, the Navier–Stokes equations seem to model fluid

motion accurately; even turbulent flows seem (on average) to agree with real world

observations.

The Navier–Stokes equations assume that the fluid being studied is a continuum

not moving at relativistic velocities. At very small scales or under extreme conditions,

real fluids made out of discrete molecules will produce results different from the

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continuous fluids modeled by the Navier–Stokes equations. Depending on the Knudsen

number of the problem, statistical mechanics or possibly even molecular dynamics may

be a more appropriate approach.

1.2.1.2 Derivation and description

The derivation of the Navier–Stokes equations begins with an application of

Newton's second law: conservation of momentum (often alongside mass and energy

conservation) being written for an arbitrary portion of the fluid. In an inertial frame of

reference, the general form of the equations of fluid motion is:

where is the flow velocity, ρ is the fluid density, p is the pressure, is the

(deviatoric) stress tensor, and represents body forces (per unit volume) acting on the

fluid and is the del operator. This is a statement of the conservation of momentum in a

fluid and it is an application of Newton's second law to a continuum; in fact this equation

is applicable to any non-relativistic continuum and is known as the Cauchy momentum

equation.

This equation is often written using the material derivative Dv/Dt, making it more

apparent that this is a statement of Newton's second law:

The left side of the equation describes acceleration, and may be composed of time

dependent or convective effects (also the effects of non-inertial coordinates if present).

The right side of the equation is in effect a summation of body forces (such as gravity)

and divergence of stress (pressure and shear stress).

A. Convective acceleration

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Fig. 1.3 An example of convection. Though the flow may be steady (time independent), the fluid

decelerates as it moves down the diverging duct (assuming incompressible flow), hence there is an

acceleration happening over position.

A very significant feature of the Navier–Stokes equations is the presence of

convective acceleration: the effect of time independent acceleration of a fluid with respect

to space. While individual fluid particles are indeed experiencing time dependent

acceleration, the convective acceleration of the flow field is a spatial effect, one example

being fluid speeding up in a nozzle. Convective acceleration is represented by the

nonlinear quantity:

which may be interpreted either as or as with the tensor

derivative of the velocity vector Both interpretations give the same result, independent

of the coordinate system provided is interpreted as the covariant derivative.

(i) Interpretation as (v·∇)v

The convection term is often written as

where the advection operator is used. Usually this representation is

preferred because it is simpler than the one in terms of the tensor derivative

(ii) Interpretation as v·(∇v)

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Here is the tensor derivative of the velocity vector, equal in Cartesian

coordinates to the component by component gradient. The convection term may, by a

vector calculus identity, be expressed without a tensor derivative:

The form has use in irrotational flow, where the curl of the velocity (called

vorticity) is equal to zero.

Regardless of what kind of fluid is being dealt with, convective acceleration is a

nonlinear effect. Convective acceleration is present in most flows (exceptions include

one-dimensional incompressible flow), but its dynamic effect is disregarded in creeping

flow (also called Stokes flow).

B. Stresses

The effect of stress in the fluid is represented by the and terms; these are

gradients of surface forces, analogous to stresses in a solid. is called the pressure

gradient and arises from the isotropic part of the stress tensor. This part is given by

normal stresses that turn up in almost all situations, dynamic or not. The anisotropic part

of the stress tensor gives rise to , which conventionally describes viscous forces; for

incompressible flow, this is only a shear effect. Thus, is the deviatoric stress tensor, and

the stress tensor is equal to:

where is the 3×3 identity matrix. Interestingly, only the gradient of pressure

matters, not the pressure itself. The effect of the pressure gradient is that fluid flows from

high pressure to low pressure.

The stress terms p and are yet unknown, so the general form of the equations of

motion is not usable to solve problems. Besides the equations of motion, Newton's second

law, a force model is needed relating the stresses to the fluid motion. For this reason,

assumptions on the specific behavior of a fluid are made (based on natural observations)

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and applied in order to specify the stresses in terms of the other flow variables, such as

velocity and density.

The Navier–Stokes equations result from the following assumptions on the deviatoric

stress tensor :

the deviatoric stress vanishes for a fluid at rest, and – by Galilean invariance –

also does not depend directly on the flow velocity itself, but only on spatial

derivatives of the flow velocity

in the Navier–Stokes equations, the deviatoric stress is expressed as the product of

the tensor gradient of the flow velocity with a viscosity tensor , i.e. :

the fluid is assumed to be isotropic, as valid for gases and simple liquids, and

consequently is an isotropic tensor; furthermore, since the deviatoric stress

tensor is symmetric, it turns out that it can be expressed in terms of two scalar

dynamic viscosities μ and μ”: where is

the rate-of-strain tensor and is the rate of expansion of the flow

the deviatoric stress tensor has zero trace, so for a three-dimensional flow

2μ + 3μ” = 0

As a result, in the Navier–Stokes equations the deviatoric stress tensor has the following

form:

with the quantity between brackets the non-isotropic part of the rate-of-strain tensor The

dynamic viscosity μ does not need to be constant – in general it depends on conditions

like temperature and pressure, and in turbulence modelling the concept of eddy viscosity

is used to approximate the average deviatoric stress.

The pressure p is modeled by use of an equation of state. For the special case of an

incompressible flow, the pressure constrains the flow in such a way that the volume of

fluid elements is constant: isochoric flow resulting in a solenoidal velocity field with

C. Other forces

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The vector field represents body forces. Typically these consist of only gravity

forces, but may include other types (such as electromagnetic forces). In a non-inertial

coordinate system, other "forces" such as that associated with rotating coordinates may be

inserted.

Often, these forces may be represented as the gradient of some scalar quantity.

Gravity in the z direction, for example, is the gradient of − ρgz. Since pressure shows up

only as a gradient, this implies that solving a problem without any such body force can be

mended to include the body force by modifying pressure.

D. Other equations

The Navier–Stokes equations are strictly a statement of the conservation of

momentum. In order to fully describe fluid flow, more information is needed (how much

depends on the assumptions made), this may include boundary data (no-slip, capillary

surface, etc.), the conservation of mass, the conservation of energy, and/or an equation of

state.

Regardless of the flow assumptions, a statement of the conservation of mass is

generally necessary. This is achieved through the mass continuity equation, given in its

most general form as:

or, using the substantive derivative:

1.2.1.3 Incompressible flow of Newtonian fluids

A simplification of the resulting flow equations is obtained when considering an

incompressible flow of a Newtonian fluid. The assumption of incompressibility rules out

the possibility of sound or shock waves to occur; so this simplification is invalid if these

phenomena are important. The incompressible flow assumption typically holds well even

when dealing with a "compressible" fluid — such as air at room temperature — at low

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Mach numbers (even when flowing up to about Mach 0.3). Taking the incompressible

flow assumption into account and assuming constant viscosity, the Navier–Stokes

equations will read, in vector form:

Here f represents "other" body forces (forces per unit volume), such as gravity or

centrifugal force. The shear stress term becomes the useful quantity ( is the

vector Laplacian) when the fluid is assumed incompressible, homogeneous and

Newtonian, where is the (constant) dynamic viscosity.

It's well worth observing the meaning of each term (compare to the Cauchy

momentum equation):

Note that only the convective terms are nonlinear for incompressible Newtonian

flow. The convective acceleration is an acceleration caused by a (possibly steady) change

in velocity over position, for example the speeding up of fluid entering a converging

nozzle. Though individual fluid particles are being accelerated and thus are under

unsteady motion, the flow field (a velocity distribution) will not necessarily be time

dependent.

Another important observation is that the viscosity is represented by the vector

Laplacian of the velocity field (interpreted here as the difference between the velocity at a

point and the mean velocity in a small volume around). This implies that Newtonian

viscosity is diffusion of momentum, this works in much the same way as the diffusion of

heat seen in the heat equation (which also involves the Laplacian).

If temperature effects are also neglected, the only "other" equation (apart from

initial/boundary conditions) needed is the mass continuity equation. Under the

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incompressible assumption, density is a constant and it follows that the equation will

simplify to:

These equations are commonly used in 3 coordinates systems: Cartesian, cylindrical,

and spherical. While the Cartesian equations seem to follow directly from the vector

equation above, the vector form of the Navier–Stokes equation involves some tensor

calculus which means that writing it in other coordinate systems is not as simple as doing

so for scalar equations (such as the heat equation).

(i) Cartesian coordinates

Writing the vector equation explicitly,

Note that gravity has been accounted for as a body force, and the values of gx, gy,

gz will depend on the orientation of gravity with respect to the chosen set of coordinates.

The continuity equation reads:

When the flow is at steady-state, ρ does not change with respect to time.

The continuity equation is reduced to:

When the flow is incompressible, ρ is constant and does not change with respect

to space. The continuity equation is reduced to:

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The velocity components (the dependent variables to be solved for) are typically

named u, v, w. This system of four equations comprises the most commonly used and

studied form. Though comparatively more compact than other representations, this is still

a nonlinear system of partial differential equations for which solutions are difficult to

obtain.

(ii) Cylindrical coordinates

A change of variables on the Cartesian equations will yield the following momentum

equations for r, φ, and z:

The gravity components will generally not be constants, however for most

applications either the coordinates are chosen so that the gravity components are constant

or else it is assumed that gravity is counteracted by a pressure field (for example, flow in

horizontal pipe is treated normally without gravity and without a vertical pressure

gradient). The continuity equation is:

This cylindrical representation of the incompressible Navier–Stokes equations is

the second most commonly seen (the first being Cartesian above). Cylindrical coordinates

are chosen to take advantage of symmetry, so that a velocity component can disappear. A

very common case is axisymmetric flow with the assumption of no tangential velocity (

uφ = 0), and the remaining quantities are independent of φ:

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(iii) Spherical coordinates

In spherical coordinates, the r, φ, and θ momentum equations are (note the convention

used: θ is polar angle, or colatitude, 0 ≤ θ ≤ π):

Mass continuity will read:

These equations could be (slightly) compacted by, for example, factoring 1 / r2 from

the viscous terms. However, doing so would undesirably alter the structure of the

Laplacian and other quantities.

(iv) Stream function formulation

Taking the curl of the Navier–Stokes equation results in the elimination of pressure.

This is especially easy to see if 2D Cartesian flow is assumed (w = 0 and no dependence

of anything on z), where the equations reduce to:

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Differentiating the first with respect to y, the second with respect to x and

subtracting the resulting equations will eliminate pressure and any conservative force.

Defining the stream function ψ through

results in mass continuity being unconditionally satisfied (given the stream

function is continuous), and then incompressible Newtonian 2D momentum and mass

conservation degrade into one equation:

where is the (2D) biharmonic operator and ν is the kinematic viscosity, .

We can also express this compactly using the Jacobian determinant:

This single equation together with appropriate boundary conditions describes 2D

fluid flow, taking only kinematic viscosity as a parameter. Note that the equation for

creeping flow results when the left side is assumed zero.

1.2.1.4 Compressible flow of Newtonian fluids

There are some phenomena that are closely linked with fluid compressibility. One

of the obvious examples is sound. Description of such phenomena requires more general

presentation of the Navier–Stokes equation that takes into account fluid compressibility.

If viscosity is assumed a constant, one additional term appears, as shown here:

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Where μv is the volume viscosity coefficient, also known as second viscosity

coefficient or bulk viscosity. This additional term disappears for an incompressible fluid,

when the divergence of the flow equals zero.

1.2.2 Euler’s equations[2]

1.2.2.1 History

The Euler equations first appeared in published form in Eulers article “Principe’s

généraux du movement des fluids,” published in Memories de l'Academie des Sciences

de Berlin in 1757. They were among the first partial differential equations to be written

down.

1.2.2.2 Conservation and component form

In differential form, the equations are:

where

ρ is the fluid mass density,

u is the fluid velocity vector, with components u, v, and w,

E = ρ e + ½ ρ ( u2 + v2 + w2 ) is the total energy per unit volume, with e being the

internal energy per unit mass for the fluid, and

p is the pressure.

The second equation includes the divergence of a dyadic product, and may be clearer

in subscript notation; for each j from 1 to 3 one has:

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where the i and j subscripts label the three Cartesian components: ( x1 , x2 , x3 ) = ( x , y , z

) and ( u1 , u2 , u3 ) = ( u , v , w ).

Note that the above equations are expressed in conservation form, as this format

emphasizes their physical origins (and is often the most convenient form for

computational fluid dynamics simulations). The second equation, which represents

momentum conservation, can also be expressed in non-conservation form as:

but this form obscures the direct connection between the Euler equations and Newton's

second law of motion.

1.2.2.3 Conservation and vector form

In vector and conservation form, the Euler equations become:

where

This form makes it clear that fx, fy and fz are fluxes.

The equations above thus represent conservation of mass, three components of

momentum, and energy. There are thus five equations and six unknowns. Closing the

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system requires an equation of state; the most commonly used is the ideal gas law (i.e.

p = ρ (γ−1) e, where ρ is the density, γ is the adiabatic index, and e the internal energy).

Note the odd form for the energy equation; see Rankine–Hugoniot equation. The

extra terms involving p may be interpreted as the mechanical work done on a fluid

element by its neighbor fluid elements. These terms sum to zero in an incompressible

fluid.

The well-known Bernoulli's equation can be derived by integrating Euler's

equation along a streamline, under the assumption of constant density and a sufficiently

stiff equation of state.

1.2.2.4 Shock waves

The Euler equations are nonlinear hyperbolic equations and their general solutions

are waves. Much like the familiar oceanic waves, waves described by the Euler Equations

'break' and so-called shock waves are formed; this is a nonlinear effect and represents the

solution becoming multi-valued. Physically this represents a breakdown of the

assumptions that led to the formulation of the differential equations, and to extract further

information from the equations we must go back to the more fundamental integral form.

Then, weak solutions are formulated by working in 'jumps' (discontinuities) into the flow

quantities – density, velocity, pressure, entropy – using the Rankine–Hugoniot shock

conditions. Physical quantities are rarely discontinuous; in real flows, these

discontinuities are smoothed out by viscosity. (See Navier–Stokes equations)

1.2.2.5 The equations in one spatial dimension

For certain problems, especially when used to analyze compressible flow in a duct

or in case the flow is cylindrically or spherically symmetric, the one-dimensional Euler

equations are a useful first approximation. Generally, the Euler equations are solved by

Riemann's method of characteristics. This involves finding curves in plane of independent

variables (i.e., x and t) along which partial differential equations (PDE's) degenerate into

ordinary differential equations (ODE's). Numerical solutions of the Euler equations rely

heavily on the method of characteristics.

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1.3 Potential flow[2]

In fluid dynamics, potential flow describes the velocity field as the gradient of a

scalar function: the velocity potential. As a result, a potential flow is characterized by an

irrotational velocity field, which is a valid approximation for several applications. The

irrotationality of a potential flow is due to the curl of a gradient always being equal to

zero.

In the case of an incompressible flow the velocity potential satisfies Laplace's

equation, and potential theory is applicable. However, potential flows also have been used

to describe compressible flows. The potential flow approach occurs in the modeling of

both stationary as well as nonstationary flows.

Applications of potential flow are for instance: the outer flow field for aerofoils,

water waves, electroosmotic flow, and groundwater flow. For flows (or parts thereof)

with strong vorticity effects, the potential flow approximation is not applicable.

1.3.1 Characteristics and applications

Fig. 1.4 Streamlines for the incompressible potential flow around a circular cylinder in a uniform on flow.

1.3.1.1 Description and characteristics

In fluid dynamics, a potential flow is described by means of a velocity potential φ,

being a function of space and time. The flow velocity v is a vector field equal to the

gradient, ∇, of the velocity potential φ:

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Sometimes, also the definition v = −∇φ, with a minus sign, is used. But here we

will use the definition above, without the minus sign. From vector calculus it is known,

that the curl of a gradient is equal to zero:

and consequently the vorticity, the curl of the velocity field v, is zero:

This implies that a potential flow is an irrotational flow. This has direct

consequences for the applicability of potential flow. In flow regions where vorticity is

known to be important, such as wakes and boundary layers, potential flow theory is not

able to provide reasonable predictions of the flow. Fortunately, there are often large

regions of a flow where the assumption of irrotationality is valid, which is why potential

flow is used for various applications. For instance in: flow around aircraft, groundwater

flow, acoustics, water waves, and electroosmotic flow.

A. Incompressible flow

In case of an incompressible flow — for instance of a liquid, or a gas at low Mach

numbers; but not for sound waves — the velocity v has zero divergence:

with the dot denoting the inner product. As a result, the velocity potential φ has to satisfy

Laplace's equation

Where is the Laplace operator (sometimes also written Δ). In this case the

flow can be determined completely from its kinematics: the assumptions of irrotationality

and zero divergence of the flow. Dynamics only have to be applied afterwards, if one is

interested in computing pressures: for instance for flow around airfoils through the use of

Bernoulli's principle.

B. Steady flow

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Potential flow theory can also be used to model irrotational compressible flow. The

full potential equation, describing a steady flow, is given by:

with Mach number components

and  

where a is the local speed of sound. The flow velocity v is again equal to ∇Φ,

with Φ the velocity potential. The full potential equation is valid for sub-, trans- and

supersonic flow at arbitrary angle of attack, as long as the assumption of irrotationality is

applicable.

In case of either subsonic or supersonic (but not transsonic or hypersonic) flow, at

small angles of attack and thin bodies, an additional assumption can be made: the velocity

potential is split into an undisturbed on flow velocity V∞ in the x-direction, and small a

perturbation velocity ∇φ thereof. So:

In that case, the linearized small-perturbation potential equation — an

approximation to the full potential equation — can be used:

with M∞ = V∞ / a∞ the Mach number of the incoming free stream. This linear equation is

much easier to solve than the full potential equation: it may be recast into Laplace's

equation by a simple coordinate stretching in the x-direction.

1.3.1.2 Derivation of full potential equation

A. Sound waves

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Small-amplitude sound waves can be approximated with the following potential-flow

model:

which is a linear wave equation for the velocity potential φ. Again the oscillatory

part of the velocity vector v is related to the velocity potential by v = ∇φ, while as before

Δ is the Laplace operator, and ā is the average speed of sound in the homogeneous

medium. Note that also the oscillatory parts of the pressure p and density ρ each

individually satisfy the wave equation, in this approximation.

1.3.1.3 Applicability and limitations

Potential flow does not include all the characteristics of flows that are encountered

in the real world. For example, potential flow excludes turbulence, which is commonly

encountered in nature. Also, potential flow theory cannot be applied for viscous internal

flows. Richard Feynman considered potential flow to be so unphysical that the only fluid

to obey the assumptions was "dry water".

More precisely, potential flow cannot account for the behavior of flows that

include a boundary layer. Nevertheless, understanding potential flow is important in

many branches of fluid mechanics.

Potential flow finds many applications in fields such as aircraft design. For

instance, in computational fluid dynamics, one technique is to couple a potential flow

solution outside the boundary layer to a solution of the boundary layer equations inside

the boundary layer.

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Chapter 2

Discretization methods[5]

Methodology

In all of these approaches the same basic procedure is followed.

During preprocessing

o The geometry (physical bounds) of the problem is defined.

o The volume occupied by the fluid is divided into discrete cells (the mesh).

The mesh may be uniform or non-uniform.

o The physical modeling is defined – for example, the equations of motions

+ enthalpy + radiation + species conservation

o Boundary conditions are defined. This involves specifying the fluid

behaviour and properties at the boundaries of the problem. For transient

problems, the initial conditions are also defined.

The simulation is started and the equations are solved iteratively as a steady-state

or transient.

Finally a postprocessor is used for the analysis and visualization of the resulting

solution.

Discretization methods

The stability of the chosen discretization is generally established numerically

rather than analytically as with simple linear problems. Special care must also be taken to

ensure that the discretization handles discontinuous solutions gracefully. The Euler

equations and Navier–Stokes equations both admit shocks, and contact surfaces.

Some of the discretization methods being used are:

2.1 Finite volume method[5]

The finite volume method is a method for representing and evaluating partial

differential equations in the form of algebraic equations [LeVeque, 2002; Toro, 1999].

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Similar to the finite difference method or finite element method, values are calculated at

discrete places on a meshed geometry. "Finite volume" refers to the small volume

surrounding each node point on a mesh. In the finite volume method, volume integrals in

a partial differential equation that contain a divergence term are converted to surface

integrals, using the divergence theorem. These terms are then evaluated as fluxes at the

surfaces of each finite volume. Because the flux entering a given volume is identical to

that leaving the adjacent volume, these methods are conservative. Another advantage of

the finite volume method is that it is easily formulated to allow for unstructured meshes.

The method is used in many computational fluid dynamics packages.

2.1.1 1D example

Consider a simple 1D advection problem defined by the following partial

differential equation

Here, represents the state variable and represents the

flux or flow of . Conventionally, positive represents flow to the right whilst negative

represents flow to the left. If we assume that equation (1) represents a flowing medium

of constant area, we can sub-divide the spatial domain, , into finite volumes or cells with

cell centres indexed as . For a particular cell, , we can define the volume average value

of at time and , as

and at time as,

where and represent locations of the upstream and downstream faces or edges

respectively of the cell.

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Integrating equation (1) in time, we have:

Where .

To obtain the volume average of at time , we integrate over

the cell volume, and divide the result by , i.e.

We assume that is well behaved and that we can reverse the order of integration.

Also, recall that flow is normal to the unit area of the cell. Now, since in one dimension

, we can apply the divergence theorem, i.e. , and substitute

for the volume integral of the divergence with the values of evaluated at the cell

surface (edges and ) of the finite volume as follows:

Where .

We can therefore derive a semi-discrete numerical scheme for the above problem

with cell centers indexed as , and with cell edge fluxes indexed as , by differentiating

(6) with respect to time to obtain:

where values for the edge fluxes, , can be reconstructed by interpolation or

extrapolation of the cell averages. Equation (7) is exact for the volume averages; i.e., no

approximations have been made during its derivation.

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2.1.2 General conservation law

We can also consider the general conservation law problem, represented by the

following PDE,

Here, represents a vector of states and represents the corresponding flux

tensor. Again we can sub-divide the spatial domain into finite volumes or cells. For a

particular cell, , we take the volume integral over the total volume of the cell, , which

gives,

On integrating the first term to get the volume average and applying the

divergence theorem to the second, this yields

where represents the total surface area of the cell and is a unit vector normal

to the surface and pointing outward. So, finally, we are able to present the general result

equivalent to (7), i.e.

Again, values for the edge fluxes can be reconstructed by interpolation or

extrapolation of the cell averages. The actual numerical scheme will depend upon

problem geometry and mesh construction.

Finite volume schemes are conservative as cell averages change through the edge

fluxes. In other words, one cell's loss is another cell's gain!

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2.2 Finite element method[2]

Fig. 2.1 2D FEM solution for a magnetostatic configuration (lines denote the direction and colour the

magnitude of calculated flux density)

Fig. 2.2 2D mesh for the image above (mesh is denser around the object of interest)

The finite element method (FEM) (its practical application often known as finite

element analysis (FEA)) is a numerical technique for finding approximate solutions of

partial differential equations (PDE) as well as of integral equations. The solution

approach is based either on eliminating the differential equation completely (steady state

problems), or rendering the PDE into an approximating system of ordinary differential

equations, which are then numerically integrated using standard techniques such as

Euler's method, Runge-Kutta, etc.

2.2.1 Application

A variety of specializations under the umbrella of the mechanical

engineering discipline (such as aeronautical, biomechanical, and automotive industries)

commonly use integrated FEM in design and development of their products. Several

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modern FEM packages include specific components such as thermal, electromagnetic,

fluid, and structural working environments. In a structural simulation, FEM helps

tremendously in producing stiffness and strength visualizations and also in minimizing

weight, materials, and costs.

FEM allows detailed visualization of where structures bend or twist, and indicates the

distribution of stresses and displacements. FEM software provides a wide range of

simulation options for controlling the complexity of both modeling and analysis of a

system. Similarly, the desired level of accuracy required and associated computational

time requirements can be managed simultaneously to address most engineering

applications. FEM allows entire designs to be constructed, refined, and optimized before

the design is manufactured.

2.2.2 Technical discussion

We will illustrate the finite element method using two sample problems from

which the general method can be extrapolated. It is assumed that the reader is familiar

with calculus and linear algebra.

P1 is a one-dimensional problem

where f is given, u is an unknown function of x, and u'' is the second derivative of u with

respect to x.

The two-dimensional sample problem is the Dirichlet problem

where Ω is a connected open region in the (x,y) plane whose boundary is "nice"

(e.g., a smooth manifold or a polygon), and uxx and uyy denote the second derivatives with

respect to x and y, respectively.

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The problem P1 can be solved "directly" by computing antiderivatives. However,

this method of solving the boundary value problem works only when there is only one

spatial dimension and does not generalize to higher-dimensional problems or to problems

like u + u'' = f. For this reason, we will develop the finite element method for P1 and

outline its generalization to P2.

Our explanation will proceed in two steps, which mirror two essential steps one must

take to solve a boundary value problem (BVP) using the FEM.

In the first step, one rephrases the original BVP in its weak form. Little to no

computation is usually required for this step. The transformation is done by hand

on paper.

The second step is the discretization, where the weak form is discretized in a finite

dimensional space.

After this second step, we have concrete formulae for a large but finite dimensional

linear problem whose solution will approximately solve the original BVP. This finite

dimensional problem is then implemented on a computer.

2.2.3 Weak formulation

The first step is to convert P1 and P2 into their equivalents weak formulation. If u solves

P1, then for any smooth function v that satisfies the displacement boundary conditions,

i.e. v = 0 at x = 0 and x = 1,we have

(1)

Conversely, if u with u(0) = u(1) = 0 satisfies (1) for every smooth function v(x) then one

may show that this u will solve P1. The proof is easier for twice continuously

differentiable u (mean value theorem), but may be proved in a distributional sense as

well.

By using integration by parts on the right-hand-side of (1), we obtain

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(2)

Where we have used the assumption that v(0) = v(1) = 0.

2.2.4 A proof outline of existence and uniqueness of the solution

We can loosely think of to be the absolutely continuous functions of

(0,1) that are 0 at x = 0 and x = 1 (see Sobolev spaces). Such function are (weakly) "once

differentiable" and it turns out that the symmetric bilinear map then defines an inner

product which turns into a Hilbert space (a detailed proof is nontrivial.) On the

other hand, the left-hand-side is also an inner product, this time on the Lp

space L2(0,1). An application of the Riesz representation theorem for Hilbert spaces

shows that there is a unique u solving (2) and therefore P1. This solution is a-priori only a

member of , but using elliptic regularity, will be smooth if f is.

2.2.5 The weak form of P2

If we integrate by parts using a form of Green's identities, we see that if u solves

P2, then for any v,

where denotes the gradient and denotes the dot product in the two-dimensional

plane. Once more can be turned into an inner product on a suitable space of

"once differentiable" functions of Ω that are zero on . We have also assumed that

(see Sobolev spaces). Existence and uniqueness of the solution can also be

shown.

2.2.6 Matrix form of the problem

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If we write and then problem (3), taking

v(x) = vj(x) for j = 1,...,n, becomes

for j = 1,...,n. (4)

If we denote by and the column vectors (u1,...,un)t and (f1,...,fn)t, and if we let

L = (Lij) and M = (Mij)

be matrices whose entries are

Lij = φ(vi,vj) and

Then we may rephrase (4) as

. (5)

It is not, in fact, necessary to assume . For a general function

f(x), problem (3) with v(x) = vj(x) for j = 1,...,n becomes actually simpler, since no matrix

M is used,

, (6)

where and for j = 1,...,n.

As we have discussed before, most of the entries of L and M are zero because the

basis functions vk have small support. So we now have to solve a linear system in the

unknown where most of the entries of the matrix L, which we need to invert, are zero.

Such matrices are known as sparse matrices, and there are efficient solvers for

such problems (much more efficient than actually inverting the matrix.) In addition, L is

symmetric and positive definite, so a technique such as the conjugate gradient method is

favored. For problems that are not too large, sparse LU decompositions and Cholesky

decompositions still work well. For instance, Matlab's backslash operator (which uses

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sparse LU, sparse Cholesky, and other factorization methods) can be sufficient for meshes

with a hundred thousand vertices.

The matrix L is usually referred to as the stiffness matrix, while the matrix M is

dubbed the mass matrix.

2.2.7 General form of the finite element method

In general, the finite element method is characterized by the following process.

One chooses a grid for Ω. In the preceding treatment, the grid consisted of

triangles, but one can also use squares or curvilinear polygons.

Then, one chooses basis functions. In our discussion, we used piecewise linear

basis functions, but it is also common to use piecewise polynomial basis

functions.

A separate consideration is the smoothness of the basis functions. For second order

elliptic boundary value problems, piecewise polynomial basis function that are merely

continuous suffice (i.e., the derivatives are discontinuous.) For higher order partial

differential equations, one must use smoother basis functions. For instance, for a fourth

order problem such as uxxxx + uyyyy = f, one may use piecewise quadratic basis functions

that are C1.

Another consideration is the relation of the finite dimensional space V to its infinite

dimensional counterpart, in the examples above . A conforming element method is

one in which the space V is a subspace of the element space for the continuous problem.

The example above is such a method. If this condition is not satisfied, we obtain a

nonconforming element method, an example of which is the space of piecewise linear

functions over the mesh which are continuous at each edge midpoint. Since these

functions are in general discontinuous along the edges, this finite dimensional space is not

a subspace of the original .

Typically, one has an algorithm for taking a given mesh and subdividing it. If the

main method for increasing precision is to subdivide the mesh, one has an h-method (h is

customarily the diameter of the largest element in the mesh.) In this manner, if one shows

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that the error with a grid h is bounded above by Chp, for some and p > 0, then one

has an order p method. Under certain hypotheses (for instance, if the domain is convex), a

piecewise polynomial of order d method will have an error of order p = d + 1.

If instead of making h smaller, one increases the degree of the polynomials used in the

basis function, one has a p-method. If one combines these two refinement types, one

obtains an hp-method (hp-FEM). In the hp-FEM, the polynomial degrees can vary from

element to element. High order methods with large uniform p are called spectral finite

element methods (SFEM). These are not to be confused with spectral methods.

For vector partial differential equations, the basis functions may take values in .

2.2.8 Various types of finite element methods

Generalized finite element method

hp-FEM

hpk-FEM

XFEM

Spectral methods

Meshfree methods

Discontinuous Galerkin methods

2.3 Finite difference method[3]

Not to be confused with "finite difference method based on variation principle",

the first name of finite element method.

In mathematics, finite-difference methods are numerical methods for

approximating the solutions to differential equations using finite difference equations to

approximate derivatives.

2.3.1 Intuitive derivation

Finite-difference methods approximate the solutions to differential equations by

replacing derivative expressions with approximately equivalent difference quotients. That

is, because the first derivative of a function f is, by definition,

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then a reasonable approximation for that derivative would be to take

for some small value of h. In fact, this is the forward difference equation for the

first derivative. Using this and similar formulae to replace derivative expressions in

differential equations, one can approximate their solutions without the need for calculus.

2.3.2 Derivation from Taylor's polynomial

Assuming the function whose derivatives are to be approximated is properly-

behaved, by Taylor's theorem,

where n! denotes the factorial of n, and Rn(x) is a remainder term, denoting the

difference between the Taylor polynomial of degree n and the original function. Again

using the first derivative of the function f as an example, by Taylor's theorem,

f(x0 + h) = f(x0) + f'(x0)h + R1(x),

\which, with some minor algebraic manipulation, is equivalent to

so that for R1(x) sufficiently small,

2.3.3 Accuracy and order

The error in a method's solution is defined as the difference between its

approximation and the exact analytical solution. The two sources of error in finite

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difference methods are round-off error, the loss of precision due to computer rounding of

decimal quantities, and truncation error or discretization error, the difference between the

exact solution of the finite difference equation and the exact quantity assuming perfect

arithmetic (that is, assuming no round-off).

Fig. 2.3 The finite difference method relies on discretizing a function on a grid.

To use a finite difference method to attempt to solve (or, more generally,

approximate the solution to) a problem, one must first discretize the problem's domain.

This is usually done by dividing the domain into a uniform grid (see image to the right).

Note that this means that finite-difference methods produce sets of discrete numerical

approximations to the derivative, often in a "time-stepping" manner.

An expression of general interest is the local truncation error of a method.

Typically expressed using Big-O notation, local truncation error refers to the error from a

single application of a method. That is, it is the quantity f'(xi) − f'i if f'(xi) refers to the

exact value and f'i to the numerical approximation. The remainder term of a Taylor

polynomial is convenient for analyzing the local truncation error. Using the Lagrange

form of the remainder from the Taylor polynomial for f(x0 + h), which is

, where x0 < ξ < x0 + h,

the dominant term of the local truncation error can be discovered. For example,

again using the forward-difference formula for the first derivative, knowing that f(xi) =

f(x0 + ih),

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and with some algebraic manipulation, this leads to

and further noting that the quantity on the left is the approximation from the finite

difference method and that the quantity on the right is the exact quantity of interest plus a

remainder, clearly that remainder is the local truncation error. A final expression of this

example and its order is:

This means that, in this case, the local truncation error is proportional to the step

size.

2.3.4 Example: ordinary differential equation

For example, consider the ordinary differential equation

The Euler method for solving this equation uses the finite difference quotient

to approximate the differential equation by first substituting in for u'(x) and applying a

little algebra to get

The last equation is a finite-difference equation, and solving this equation gives an

approximate solution to the differential equation.

2.3.5 Example: The heat equation

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Consider the normalized heat equation in one dimension, with homogeneous

Dirichlet boundary conditions

( boundary condition)

( initial condition)

One way to numerically solve this equation is to approximate all the derivatives

by finite differences. We partition the domain in space using a mesh x0,...,xJ and in time

using a mesh t0,....,tN. We assume a uniform partition both in space and in time, so the

difference between two consecutive space points will be h and between two consecutive

time points will be k. The points

will represent the numerical approximation of u(xj,tn).

2.3.6 Explicit method

Fig. 2.4 The stencil for the most common explicit method for the heat equation.

Using a forward difference at time tn and a second-order central difference for the

space derivative at position xj ("FTCS") we get the recurrence equation:

This is an explicit method for solving the one-dimensional heat equation.

We can obtain from the other values this way:

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where r = k / h2.

So, with this recurrence relation, and knowing the values at time n, one can obtain

the corresponding values at time n+1. and must be replaced by the boundary

conditions, in this example they are both 0.

This explicit method is known to be numerically stable and convergent whenever

. The numerical errors are proportional to the time step and the square of the space

step:

2.3.7 Implicit method

Fig. 2.5 The implicit method stencil.

If we use the backward difference at time tn + 1 and a second-order central

difference for the space derivative at position xj ("BTCS") we get the recurrence equation:

This is an implicit method for solving the one-dimensional heat equation.

We can obtain from solving a system of linear equations:

The scheme is always numerically stable and convergent but usually more

numerically intensive than the explicit method as it requires solving a system of

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numerical equations on each time step. The errors are linear over the time step and

quadratic over the space step.

2.3.8 Crank–Nicolson method

Finally if we use the central difference at time tn + 1 / 2 and a second-order central

difference for the space derivative at position xj ("CTCS") we get the recurrence equation:

This formula is known as the Crank–Nicolson method.

Fig. 2.6 The Crank–Nicolson stencil.

We can obtain from solving a system of linear equations:

The scheme is always numerically stable and convergent but usually more

numerically intensive as it requires solving a system of numerical equations on each time

step. The errors are quadratic over the time step and formally are of the fourth degree

regarding the space step:

However, near the boundaries, the error is often O(h2) instead of O(h4).

Usually the Crank–Nicolson scheme is the most accurate scheme for small time

steps. The explicit scheme is the least accurate and can be unstable, but is also the easiest

to implement and the least numerically intensive. The implicit scheme works the best for

large time steps.

2.4 Boundary element method[5]

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The boundary element method (BEM) is a numerical computational method of

solving linear partial differential equations which have been formulated as integral

equations (i.e. in boundary integral form). It can be applied in many areas of engineering

and science including fluid mechanics, acoustics, electromagnetics, and fracture

mechanics. (In electromagnetics, the more traditional term "method of moments" is often,

though not always, synonymous with "boundary element method".)

2.4.1 Mathematical basis

The integral equation may be regarded as an exact solution of the governing

partial differential equation. The boundary element method attempts to use the given

boundary conditions to fit boundary values into the integral equation, rather than values

throughout the space defined by a partial differential equation. Once this is done, in the

post-processing stage, the integral equation can then be used again to calculate

numerically the solution directly at any desired point in the interior of the solution

domain.

2.5 High-resolution scheme[2]

High-resolution schemes are used in the numerical solution of partial differential

equations where high accuracy is required in the presence of shocks or discontinuities.

They have the following properties:

Second or higher order spatial accuracy is obtained in smooth parts of the

solution.

Solutions are free from spurious oscillations or wiggles.

High accuracy is obtained around shocks and discontinuities.

The number of mesh points containing the wave is small compared with a first-

order scheme with similar accuracy.

High-resolution schemes often use flux/slope limiters to limit the gradient around

shocks or discontinuities. A particularly successful high-resolution scheme is the MUSCL

scheme which uses state extrapolation and limiters to achieve good accuracy.

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Chapter 3

Turbulence models[2]

This section requires expansion with:

Models should be listed in some order (increasing cost, increasing accuracy,

alphabetically...)

Add the One-dimensional turbulence model (and its predecessor Linear

eddy model).

In studying turbulent flows, the objective is to obtain a theory or a model that can

yield quantities of interest, such as velocities. For turbulent flow, the range of length

scales and complexity of phenomena make most approaches impossible. The primary

approach in this case is to create numerical models to calculate the properties of interest.

A selection of some commonly-used computational models for turbulent flows are

presented in this section.

The chief difficulty in modeling turbulent flows comes from the wide range of

length and time scales associated with turbulent flow. As a result, turbulence models can

be classified based on the range of these length and time scales that are modeled and the

range of length and time scales that are resolved. The more turbulent scales that are

resolved, the finer the resolution of the simulation, and therefore the higher the

computational cost. If a majority or all of the turbulent scales are modeled, the

computational cost is very low, but the tradeoff comes in the form of decreased accuracy.

3.1 Direct numerical simulation[4]

A direct numerical simulation (DNS) is a simulation in computational fluid

dynamics in which the Navier-Stokes equations are numerically solved without any

turbulence model. This means that the whole range of spatial and temporal scales of the

turbulence must be resolved. All the spatial scales of the turbulence must be resolved in

the computational mesh, from the smallest dissipative scales (Kolmogorov microscales),

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up to the integral scale L, associated with the motions containing most of the kinetic

energy. The Kolmogorov scale,η, is given by

where ν is the kinematic viscosity and ε is the rate of kinetic energy dissipation.

On the other hand, the integral scale depends usually on the spatial scale of the boundary

conditions.

To satisfy these resolution requirements, the number N of points along a given

mesh direction with increments h, must be

,

So that the integral scale is contained within the computational domain, and also

,

So that the Kolmogorov scale can be resolved.

Since

,

Where u' is the root mean square (RMS) of the velocity, the previous relations imply that

a three-dimensional DNS requires a number of mesh points N3 satisfying

Where Re is the turbulent Reynolds number:

.

Hence, the memory storage requirement in a DNS grows very fast with the

Reynolds number. In addition, given the very large memory necessary, the integration of

the solution in time must be done by an explicit method. This means that in order to be

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accurate, the integration must be done with a time step, Δt, small enough such that a fluid

particle moves only a fraction of the mesh spacing h in each step. That is,

(C is here the Courant number). The total time interval simulated is generally

proportional to the turbulence time scale τ given by

.

Combining these relations, and the fact that h must be of the order of η, the

number of time-integration steps must be proportional to L / (Cη). By other hand, from

the definitions for Re, η and L given above, it follows that

,

and consequently, the number of time steps grows also as a power law of the

Reynolds number.

One can estimate that the number of floating-point operations required to

complete the simulation is proportional to the number of mesh points and the number of

time steps, and in conclusion, the number of operations grows as Re3.

Therefore, the computational cost of DNS is very high, even at low Reynolds

numbers. For the Reynolds numbers encountered in most industrial applications, the

computational resources required by a DNS would exceed the capacity of the most

powerful computers currently available. However, direct numerical simulation is a useful

tool in fundamental research in turbulence. Using DNS it is possible to perform

"numerical experiments", and extract from them information difficult or impossible to

obtain in the laboratory, allowing a better understanding of the physics of turbulence.

Also, direct numerical simulations are useful in the development of turbulence models for

practical applications, such as sub-grid scale models for Large eddy simulation (LES) and

models for methods that solve the Reynolds-averaged Navier-Stokes equations (RANS).

This is done by means of "a priori" tests, in which the input data for the model is taken

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from a DNS simulation, or by "a posteriori" tests, in which the results produced by the

model are compared with those obtained by DNS. The biggest DNS in the world, up to

this date, used 40963 mesh points. It was carried out in the Japanese Earth Simulator

supercomputer in 2002.

3.2 Large eddy simulation[2]

Fig. 3.1 Large eddy simulation of a turbulent gas velocity field.

Large eddy simulation (LES) is a mathematical model for turbulence used in

computational fluid dynamics. It was initially proposed in 1963 by Joseph Smagorinsky

to simulate atmospheric air currents, and many of the issues unique to LES were first

explored by Deardorff (1970). LES grew rapidly and is currently applied in a wide variety

of engineering applications, including combustion, acoustics, and simulations of the

atmospheric boundary layer. LES operates on the Navier-Stokes equations to reduce the

range of length scales of the solution, reducing the computational cost.

The principal operation in large eddy simulation is low-pass filtering. This

operation is applied to the Navier-Stokes equations to eliminate small scales of the

solution. This reduces the computational cost of the simulation. The governing equations

are thus transformed, and the solution is a filtered velocity field. Which of the "small"

length and time scales to eliminate are selected according to turbulence theory and

available computational resources.

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Large eddy simulation resolves large scales of the flow field solution, allowing

better fidelity than alternative approaches that do not resolve any scales of the solution

(such as Reynolds-averaged Navier-Stokes (RANS) methods). It also models the smallest

(and most expensive) scales of the solution, rather than resolving them. This makes the

computational cost for practical engineering systems with complex geometry or flow

configurations, such as turbulent jets, pumps, vehicles, and landing gear, attainable using

supercomputers. In contrast, direct numerical simulation, which resolves every scale of

the solution, is prohibitively expensive for nearly all systems with complex geometry or

flow configurations.

3.3 Detached eddy simulation[2]

Detached eddy simulation (DES) is a modification of a RANS model in which the

model switches to a subgrid scale formulation in regions fine enough for LES

calculations. Regions near solid boundaries and where the turbulent length scale is less

than the maximum grid dimension are assigned the RANS mode of solution. As the

turbulent length scale exceeds the grid dimension, the regions are solved using the LES

mode. Therefore the grid resolution is not as demanding as pure LES, thereby

considerably cutting down the cost of the computation. Though DES was initially

formulated for the Spalart-Allmaras model, it can be implemented with other RANS

models (Strelets, 2001), by appropriately modifying the length scale which is explicitly or

implicitly involved in the RANS model.

3.4 Reynolds-averaged Navier–Stokes[2]

Reynolds-averaged Navier-Stokes (RANS) equations are the oldest approach to

turbulence modeling. An ensemble version of the governing equations is solved, which

introduces new apparent stresses known as Reynolds stresses. This adds a second order

tensor of unknowns for which various models can provide different levels of closure. It is

a common misconception that the RANS equations do not apply to flows with a time-

varying mean flow because these equations are 'time-averaged'. In fact, statistically

unsteady (or non-stationary) flows can equally be treated. This is sometimes referred to as

URANS. There is nothing inherent in Reynolds averaging to preclude this, but the

turbulence models used to close the equations are valid only as long as the time over

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which these changes in the mean occur is large compared to the time scales of the

turbulent motion containing most of the energy.

RANS models can be divided into two broad approaches:

Boussinesq hypothesis

This method involves using an algebraic equation for the Reynolds stresses which include

determining the turbulent viscosity, and depending on the level of sophistication of the

model, solving transport equations for determining the turbulent kinetic energy and

dissipation. Models include k-ε (Spaldin), Mixing Length Model (Prandtl) and Zero

Equation (Chen). The models available in this approach are often referred to by the

number of transport equations associated with the method. For example, the Mixing

Length model is a "Zero Equation" model because no transport equations are solved; the k

− ε is a "Two Equation" model because two transport equations (one for k and one for ε)

are solved.

Reynolds stress model (RSM)

This approach attempts to actually solve transport equations for the Reynolds stresses.

This means introduction of several transport equations for all the Reynolds stresses and

hence this approach is much more costly in CPU effort.

3.5 Coherent vortex simulation[2]

The coherent vortex simulation approach decomposes the turbulent flow field into

a coherent part, consisting of organized vortical motion, and the incoherent part, which is

the random background flow. This decomposition is done using wavelet filtering. The

approach has much in common with LES, since it uses decomposition and resolves only

the filtered portion, but different in that it does not use a linear, low-pass filter. Instead,

the filtering operation is based on wavelets, and the filter can be adapted as the flow field

evolves. Farge and Schneider tested the CVS method with two flow configurations and

showed that the coherent portion of the flow exhibited the energy spectrum exhibited by

the total flow, and corresponded to coherent structures (vortex tubes), while the

incoherent parts of the flow composed homogeneous background noise, which exhibited

no organized structures. Goldstein and Oleg applied the CVS model to large eddy

simulation, but did not assume that the wavelet filter completely eliminated all coherent

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motions from the subfilter scales. By employing both LES and CVS filtering, they

showed that the SFS dissipation was dominated by the SFS flow field's coherent portion.

3.6 PDF methods[2]

Probability density function (PDF) methods for turbulence, first introduced by

Lundgren, are based on tracking the one-point PDF of the velocity, , which gives the

probability of the velocity at point being between and . This approach is analogous to the

kinetic theory of gases, in which the macroscopic properties of a gas are described by a

large number of particles. PDF methods are unique in that they can be applied in the

framework of a number of different turbulence models; the main differences occur in the

form of the PDF transport equation. For example, in the context of large eddy simulation,

the PDF becomes the filtered PDF. PDF methods can also be used to describe chemical

reactions, and are particularly useful for simulating chemically reacting flows because the

chemical source term is closed and does not require a model. The PDF is commonly

tracked by using Lagrangian particle methods; when combined with large eddy

simulation, this leads to a Langevin equation for subfiler particle evolution.

3.7 Vortex method[2]

The vortex method is a grid-free technique for the simulation of turbulent flows. It

uses vortices as the computational elements, mimicking the physical structures in

turbulence. Vortex methods were developed as a grid-free methodology that would not be

limited by the fundamental smoothing effects associated with grid-based methods. To be

practical, however, vortex methods require means for rapidly computing velocities from

the vortex elements – in other words they require the solution to a particular form of the

N-body problem (in which the motion of N objects is tied to their mutual influences).

Software based on the vortex method offer a new means for solving tough fluid

dynamics problems with minimal user intervention. All that is required is specification of

problem geometry and setting of boundary and initial conditions. Among the significant

advantages of this modern technology;

It is practically grid-free, thus eliminating numerous iterations associated with

RANS and LES.

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All problems are treated identically. No modeling or calibration inputs are

required.

Time-series simulations, which are crucial for correct analysis of acoustics, are

possible.

The small scale and large scale are accurately simulated at the same time.

3.8 Vorticity confinement method[2]

The vorticity confinement (VC) method is an Eulerian technique used in the

simulation of turbulent wakes. It uses a solitary-wave like approach to produce stable

solution with no numerical spreading. VC can capture the small scale features to over as

few as 2 grid cells. Within these features, a nonlinear difference equation is solved as

opposed to the finite difference equation. VC is similar to shock capturing methods,

where conservation laws are satisfied, so that the essential integral quantities are

accurately computed.

Two-phase flow[2]

The modeling of two-phase flow is still under development. Different methods

have been proposed.The Volume of fluid method has received a lot of attention lately, for

problems that do not have dispersed particles, but the Level set method and front tracking

are also valuable approaches. Most of these methods are either good in maintaining a

sharp interface or at conserving mass. This is crucial since the evaluation of the density,

viscosity and surface tension is based on the values averaged over the interface.

Lagrangian multiphase models, which are used for dispersed media, are based on solving

the Lagrangian equation of motion for the dispersed phase.

Solution algorithms

Discretization in space produces a system of ordinary differential equations for unsteady

problems and algebraic equations for steady problems. Implicit or semi-implicit methods

are generally used to integrate the ordinary differential equations, producing a system of

(usually) nonlinear algebraic equations. Applying a Newton or Picard iteration produces a

system of linear equations which is nonsymmetric in the presence of advection and

indefinite in the presence of incompressibility. Such systems, particularly in 3D, are

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frequently too large for direct solvers, so iterative methods are used, either stationary

methods such as successive overrelaxation or Krylov subspace methods. Krylov methods

such as GMRES, typically used with preconditioning, operate by minimizing the residual

over successive subspaces generated by the preconditioned operator.

Multigrid has the advantage of asymptotically optimal performance on many

problems. Traditional solvers and preconditioners are effective at reducing high-frequency

components of the residual, but low-frequency components typically require many

iterations to reduce. By operating on multiple scales, multigrid reduces all components of

the residual by similar factors, leading to a mesh-independent number of iterations.

For indefinite systems, preconditioners such as incomplete LU factorization,

additive Schwarz, and multigrid perform poorly or fail entirely, so the problem structure

must be used for effective preconditioning. Methods commonly used in CFD are the

SIMPLE and Uzawa algorithms which exhibit mesh-dependent convergence rates, but

recent advances based on block LU factorization combined with multigrid for the

resulting.

Chapter 4

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Conclusion

Computational Fluid Dynamics (CFD) is the science of determining a numerical

solution to the governing equations of fluid flow whilst advancing the solution through

space and time to obtain a numerical description of the complete flow field of interest. As

a developing science, CFD has received extensive attention throughout the international

community since the advent of the digital computer.

The attraction of the subject is twofold. Firstly, the desire to be able to model

physical fluid phenomena that cannot be easily simulated or measured with a physical

experiment, for example weather systems or hypersonic aerospace vehicles. Secondly, the

desire to be able to investigate physical fluid systems more cost effectively and more

rapidly than with experimental procedures.

By use of the CFD softwares we are able to simulate the very difficult problems of

fluid flow very effectively that is not possible by conventional methods of solving

problem of fluid flow. This branch of fluid mechanics will also find the solutions and

simulation techniques for two phase flow in near future which will be big achievement for

all of us.

Bibliography

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1. Patankar, Suhas (1980), Numerical Heat Transfer and Fluid Flow, Hemisphere

Series on Computational Methods in Mechanics and Thermal Science, Taylor &

Francis, ISBN 08911652239.

2. www.wikipedia.org

3. Anderson, John D. (1995), Computational Fluid Dynamics: The Basics With

Applications, McGraw-Hill Science, ISBN 0070016852

4. www.cfd-online.com

5. Sayma, Abdulnaser, (2009), Computational Fluid Dynamics, Vntus Publishing, ISBN 9788776814304.

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Contents

Chapter Page no.

1. COMPUTATIONAL FLUID DYNAMICS

1.1. INTRODUCTION 1

1.1.1. BACKGROUND AND HISTORY 1

1.2. BASIC EQUATION USED TO DETERMINING THE FLUID FLOW 2

1.2.1. NAVIER-STOKES EQUATIONS 2

1.2.1.1. PROPERTIES 3

A. NONLINEARITY 3

B. TURBULENCE 3

C. APPLICABILITY 3

1.2.1.2. DERIVATION AND DESCRIPTION 3

A. CONVECTIVE ACCELERATION 4

(i) INTERPRETATION AS (V.▼)V 4

(ii) INTERPRETATION AS V.(▼V) 5

B. STRESSES 5

C. OTHER FORCES 6

D. OTHER EQUATIONS 8

1.2.1.3. INCOMPRESSIBLE FLOW OF NEWTONIAN FLUIDS 8

(i) CARTESIAN COORDINATES 8

(ii) CYLINDRICAL COORDINATES 10

(iii) SPHERICAL COORDINATES 11

(iv) STREAM FUNCTION FORMULATION 12

1.2.1.4. COMPRESSIBLE FLOW OF NEWTONIAN FLUIDS 12

1.2.2. EULER’S EQUATIONS 13

1.2.2.1. HISTORY 14

1.2.2.2. CONSERVATION AND COMPONENT FORM 14

1.2.2.3. CONSERVATION AND VECTOR FORM 14

1.2.2.4. SHOCK WAVES 16

1.2.2.5. THE EQUATION OF ONE SPATIAL DIMENSION 16

1.3. POTENTIAL FLOW 17

1.3.1. CHARACTERISTICS AND APPLICATIONS 17

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1.3.1.1. DESCRIPTION AND CHARACTERISTICS 17

A. INCOMPRESSIBLE FLOW 18

B. STEADY FLOW 19

1.3.1.2. DERIVATION OF FULL POTENTIAL EQUATION 20

A. SOUND WAVES 20

1.3.1.3 APPLICABILITY AND LIMITATIONS 20

2. DISCRETIZATION METHODS

METHODOLOGY 21

DISCRETIZATION METHODS 21

2.1 FINITE VOLUME METHOD 21

2.1.1 1D EXAMPLE 22

2.1.2 GENERAL CONVERSION LAW 24

2.2 FINITE ELEMENT METHOD 25

2.2.1 APPLICATION 25

2.2.2 TECHNICAL DISCUSSION 26

2.2.3 WEAK FORMULATION 27

2.2.4 A PROOF OUTLINE OF EXISTENCE AND UNIQUENESS OF THE

SOLUTION 28

2.2.5 THE WEAK FORM OF P2 28

2.2.6 MATRIX FORM OF THE PROBLEM 28

2.2.7 GENERAL FORM OF FINITE ELEMENT METHOD 30

2.2.8 VARIOUS TYPE OF FINITE ELEMENT METHODS 31

2.3 FINITE DIFFERENCE METHOD 31

2.3.1 INTUITIVE DERIVATION 31

2.3.2 DERIVATION OF TAYLOR’S POLYNOMIAL 32

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2.3.3 ACCURACY AND ORDER 32

2.3.4 EXAMPLE: ORDINARY DIFFERENTIAL EQUATION 34

2.3.5 EXAMPLE: THE HEAT EQUATION 34

2.3.6 EXPLICIT METHOD 35

2.3.7 IMPLICIT METHOD 36

2.3.8 CRANK-NICOLSON METHOD 37

2.4 BOUNDARY ELEMENT METHOD 37

2.4.1 MATHEMATICAL BASIS 38

2.5 HIGH RESOLUTION SCHEME 38

3. TURBULENCE MODELS

3.1 DIRECT NUMERICAL SIMULATION 39

3.2 LARGE EDDY SIMULATION 42

3.3 DETACHED EDDY SIMULATION 43

3.4 RAYNOLDS- AVERAGED NAVIER-STOKES 43

3.5 COHERENT VORTEX SIMULATION 44

3.6 PDF METHODS 45

3.7 VORTEX METHOD 45

3.8 VORTEX CONFINEMENT METHOD 46

TWO PHASE FLOW 46

SOLUTION ALGORITHMS 46

4. CONCLUSION 48

5. BIBLIOGRAPHY 49

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LIST OF FIGURE

Figure Page no.

1.1 A computer simulation of high velocity air flow around the Space Shuttle during re

entry. 1

1.2 A simulation of the Hyper-X scramjet vehicle in operation at Mach-7 1

1.3 An example of convection. 5

1.4 Streamlines for the incompressible potential flow around a circular cylinder in a

uniform on flow. 22

2.1 2D FEM solution for a magnetostatic configuration 25

2.2 2D mesh for the image above. 25

2.3 The finite difference method relies on discretizing a function on a grid. 33

2.4 The stencil for the most common explicit method for the heat equation. 35

2.5 The implicit method stencil. 36

2.6 The Crank-Nicolson method stencil. 37

3.1 Large eddy simulation of a turbulent gas velocity field. 42

ACKNOWLEDGEMENT

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Any mission can never conclude without cooperation from surrounding. It is a great

pleasure for me to make use of this opportunity to express my thanks those person who

help me to bring out my seminar a successful venture.

First of all, I express my deep sense of gratitude and sincere thanks to my seminar guide,

“Mr. D P Sharma”, for encouraging me to deliver my seminar in the topic of my interest

and also given me the constant invaluable guidance, in the absence of which my seminar

presentation would have been incomplete.

I would also like to express my sincere thanks to “Dr. V K Gorana”, and “Mr. N K

Gupta”, for kindly providing me an opportunity to present the seminar.

Last but not the least, I thank all of my colleagues and friends for motivating me

throughout the completion of my seminar report.

Naresh Kumar,

07/391

Final Year, Mech.

UCE, Kota.

A

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Seminar ReportOn

“Computational Fluid Dynamics”

Submitted in partial fulfillment of the requirementFor the award of the

Degree ofBachelor of Technology

InMechanical Engineering

Submitted by Under the supervision ofNaresh Kumar Mr. D P SharmaFinal B.Tech. Asst. professor07/391 UCE, Kota

Department of Mechanical Engineering

RAJASTHAN TECHNICAL UNIVERSITYKOTA

CERTIFICATE

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This is to certify that Mr Naresh Kumar a student of B.Tech. (Mechanical Engineering) VIIIth semester has submitted his seminar entitled “Computational Fluid Dynamics” under my guidance.

Seminar GuideMr. D P SharmaAsst. professorDeptt. of MechanicalEngg.,UCE, Kota

ABSTRACT

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Fluid (gas and liquid) flows are governed by partial differential equations which

represent conservation laws for the mass, momentum, and energy.

Computational Fluid Dynamics (CFD) is the art of replacing such PDE systems

by a set of algebraic equations which can be solved using digital computers.

Fluid flows encountered in everyday life include meteorological phenomena (rain, wind,

hurricanes, floods, fires), environmental hazards (air pollution, transport of

contaminants), heating, ventilation and air conditioning of buildings, cars etc.,

combustion in automobile engines and other propulsion systems, interaction of various

objects with the surrounding air/water, complex flows in furnaces, heat exchangers,

chemical reactors etc., processes in human body (blood flow, breathing, drinking . . . )

and so on and so forth.

So we need various calculations to be performed, testing of flow widh the help of wind

tunnel or anything simulated environment for exact solution of flow problems. These

problems can be solved by using the branch of fluid mechanics which use numerical

methods, qualitative predictions of fluid flow.Nowdays these calculations and

computation is done by CFD softwares.

Computational Fluid Dynamics (CFD) provides a qualitative (and sometimes even

quantitative) prediction of fluid flows by means of

• mathematical modeling (partial differential equations)

• numerical methods (discretization and solution techniques)

• software tools (solvers, pre- and postprocessing utilities)

CFD enables scientists and engineers to perform ‘numerical experiments’

(i.e. computer simulations) in a ‘virtual flow laboratory’

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