NGM_JF006_1: Computational Fluid Dynamics Széchenyi University Instructor: D. Feszty, T. Jakubík Audi Hungaria Dept. of Whole Vehicle Development _____________________________________________________________________________________ 1 1. FLUID MECHANICS BASICS: A BRIEF REVIEW Fluid Mechanics is the science dealing with the action of forces on fluids. Fluids are substances, which flow freely. This is because particles in the fluid can easily move and change their relative position. See the PowerPoint slideshow called “FluidsI_Introduction.pdf” for examples on application of Fluid Mechanics in mechanical, aerospace, biomedical, renewable energy, civil engineering. 1.1. DEFINITIONS 1.1.1. Distinction between solids, liquids, gases It is the molecular structure which distinguishes solids, liquids and gases: Tab. 1. Properties of solids. liquids, fluids Solids Liquids Gases Molecular structure Lattice formation None None Molecular motion No motion Free motion Free motion Molecular spacing Constant & small Constant & small Varying & large Volume Constant: defined by molecular spacing Constant: defined by molecular spacing Any: gas completely fills space Volume defined Volume defined Any volume
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1. FLUID MECHANICS BASICS: A BRIEF REVIEW...1 1. FLUID MECHANICS BASICS: A BRIEF REVIEW Fluid Mechanics is the science dealing with the action of forces on fluids. Fluids are substances,
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NGM_JF006_1: Computational Fluid Dynamics Széchenyi University Instructor: D. Feszty, T. Jakubík Audi Hungaria Dept. of Whole Vehicle Development
Pressure is basically a specific type of stress. The general definition of stress is:
Force Stress = ------------------------------------ N/m2 or Pa Area over which force acts Depending on the direction of the force, we can distinguish between: Shear stress:
𝜏 =𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑖𝑎𝑙 𝑓𝑜𝑟𝑐𝑒
𝑎𝑟𝑒𝑎=
𝐹𝑇
𝐴
Normal stress:
𝜎 =𝑛𝑜𝑟𝑚𝑎𝑙 𝑓𝑜𝑟𝑐𝑒
𝑎𝑟𝑒𝑎=
𝐹𝑁
𝐴
NGM_JF006_1: Computational Fluid Dynamics Széchenyi University Instructor: D. Feszty, T. Jakubík Audi Hungaria Dept. of Whole Vehicle Development
SHEAR STRESS NORMAL STRESS However, since fluids cannot sustain tensible stress, only the compressive stress can be considered for them, which is in fact called “pressure”: pressure = compressive stress
𝑝 =𝑛𝑜𝑟𝑚𝑎𝑙 𝑓𝑜𝑟𝑐𝑒
𝑎𝑟𝑒𝑎=
𝐹𝑁
𝐴
Moreover, fluids cannot sustain shear stress without motion, i.e. whenever shear stress is applied to fluids, motion occurs. This is the basic difference between FLUIDS and SOLIDS.
SOLID FLUID
NGM_JF006_1: Computational Fluid Dynamics Széchenyi University Instructor: D. Feszty, T. Jakubík Audi Hungaria Dept. of Whole Vehicle Development
A FLUID is a substance, which cannot sustain shear stress statically, i.e. without motion.
1.2.5. The no-slip condition Notice that in the figure above, for a liquid point “A” remained stationary on the surface above and it is only point “B”, which moves due to the application of shear stress. This is due to the fact, that at solid boundaries the fluid and solid molecules “interlock” and there is no relative motion between them. This is called the “no-slip condition” in Fluid Mechanics.
The no-slip condition always occurs, no matter how “smooth” the solid surface is (there will always be some finite surface roughness). In other words, the no-slip condition means that in a real fluid the velocity at the wall is always zero. Note that often we will consider idealized fluids, in which we will neglect the no-slip condition. 1.2.6. Viscosity In plain terms, viscosity is essentially the measure of how “sticky” (or viscous) a fluid is. For example, honey is more viscous than water and water is more viscous than air. The scientific explanation of viscosity is the following:
When shear stress () is applied to a fluid, there will be a “velocity profile” in the fluid.
NGM_JF006_1: Computational Fluid Dynamics Széchenyi University Instructor: D. Feszty, T. Jakubík Audi Hungaria Dept. of Whole Vehicle Development
The velocity profile is proportional (denoted by the ≈ sign) to the shear stress () as:
𝑣𝐵 − 𝑣𝐴
∆𝑦 ≈ 𝜏
𝑑𝑣
𝑑𝑦≈ 𝜏
And the constant of proportionality is nothing else than viscosity () pronounced as “mu”.
𝜇𝑑𝑣
𝑑𝑦≈ 𝜏
There are two types of viscosity defined in Fluid Mechanics:
Dynamic (or absolute) viscosity: units in [N.s/m2]
Kinematic viscosity: 𝜗 = 𝜇
𝜌 units in [m2/s]
Note: the upper definition is called “dynamic” viscosity, because a force term (the Newton) occurs in the units. Recall from Sec. 1.4. that when we call something “dynamic” that means that we consider the forces acting on the object. In real life, the velocity profile v = f(y) is not linear as shown in the sketches above, but usually non-linear:
and this is called the “boundary layer” of the fluid
NGM_JF006_1: Computational Fluid Dynamics Széchenyi University Instructor: D. Feszty, T. Jakubík Audi Hungaria Dept. of Whole Vehicle Development
In real life, all object on the ground, let them be stationary (e.g. buildings, skyscrapers, stadiums, wind turbines, etc.) or moving (cars, aircraft, trains, etc.) lie in the boundary layer of the Earth, when wind is blowing. A moving car or aircraft creates its own boundary layer around the surface. The boundary layer thickness around a moving car or aircraft is in the order of 5-20 mm, whereas the boundary layer thickness of Earth (due to wind blowing) can be as much as 100 m high. This is one reason why wind turbines are so tall: to place the actual rotors as high as possible, where the wind velocity is higher than close to the ground. 1.3. FLUIDS IN MOTION
In Fluid Kinematics or Fluid Dynamics, the knowledge of the velocity distribution throughout the
flowfield is imperative for determining the forces acting on the body immersed in the fluid.
For example, to determine the lift generated by an airplane wing, we need to know the velocity
distribution around it. Or, to determine the wind loads acting on a skyscraper or a wind turbine,
we need to know the velocity distribution along the building or the turbine blades.
In Fluid Mechanics, there are two ways to express the equations for fluids in motion: in the
Lagrangian or in the Eulerian viewpoint.
1.3.1. Lagrangian viewpoint
In this approach, individual fluid particles are considered and their motion “tracked”:
Position of fluid particle: 𝑟(𝑡) = 𝑥𝑖 + 𝑦𝑗 + 𝑧�⃗⃗�
with: x, y, z – coordinates
𝑖,𝑗, �⃗⃗� – unit vectors
Velocity of fluid particle: �⃗�(𝑡) =𝑑𝑟
𝑑𝑡=
𝑑𝑥
𝑑𝑡𝑖 +
𝑑𝑦
𝑑𝑡𝑗 +
𝑑𝑧
𝑑𝑡�⃗⃗�
with: u, v, w – velocity components in x, y, z coordinates
NGM_JF006_1: Computational Fluid Dynamics Széchenyi University Instructor: D. Feszty, T. Jakubík Audi Hungaria Dept. of Whole Vehicle Development
To describe the entire flowfield, each and every particle has to be considered simultaneously. Good for rarefied gas dynamics (i.e. edge of space), where the number of particles is relatively small.
We don’t distinguish between individual particles. Instead, we consider their total effect in an elementary small volume (such as point P in space). Good for fluids, where the number of particles is large.
The above expressions are for one (1) individual particle only. For the entire flowfield, the
motion of ALL fluid particles has to be considered simultaneously. This is usually a very
challenging task for large number of particles, such as air at sea level or water. Hence, we
usually prefer to use the second approach…
1.3.2. Eulerian viewpoint
In this approach, we are focussing on a certain point in space and consider the motion of fluid
particles passing that point as time goes by.
Position in space: P[x, y, z]
Velocity at point P: u = f1(x, y, z, t)
v = f2(x, y, z, t)
w = f3(x, y, z, t)
1.3.3. Graphical illustration
The difference between the Lagrangian and Eulerian viewpoints can be illustrated graphically
as:
LAGRANGIAN EULERIAN
particle 1
particle 2
elementary small volume
inflow
w
outflow
w
dx
dy
dz
NGM_JF006_1: Computational Fluid Dynamics Széchenyi University Instructor: D. Feszty, T. Jakubík Audi Hungaria Dept. of Whole Vehicle Development
NON-UNIFORM FLOW: change of velocity along the streamline
(typically means non-parallel or non-straight streamlines)
𝜕�⃗⃗�
𝜕𝑠≠ 0
converging flow vortex flow (change of velocity (change of velocity magnitude along direction along
streamline) streamline) 1.3.6. Steady vs. unsteady flow (variation of flow in time) STEADY FLOW: at any given point, the velocity does NOT vary with time in magnitude or
direction. Example: airfoil at low angle of attack.
𝜕�⃗�
𝜕𝑡= 0
Computational streaklines (left) and smoke traces in a wind tunnel (right) around an airfoil. Source: hu.wikipedia.org & NASA
NGM_JF006_1: Computational Fluid Dynamics Széchenyi University Instructor: D. Feszty, T. Jakubík Audi Hungaria Dept. of Whole Vehicle Development
1.3.7. Laminar vs. turbulent flow LAMINAR FLOW: a very “organized” flow, in which fluid “layers” shear on each other. TURBULENT FLOW: characterised by mixing action throughout the flowfield, caused by eddies
of varying size within the flow. Turbulent flow is essentially an unsteady flow but due to the small-scale
of eddies we usually look at average effects and so we consider it steady flow.
1.3.8. 1D, 2D and 3D flow The most general representation of a fluid particle’s velocity is by 3 components: [u, v, w]. 1D flow: flow conditions change along one dimension only and are constant over
2D flow: flow conditions change along 2 dimensions only (third component of velocity is zero) flow in this plane is essentially 2D infinitely long wing 3D flow: flow conditions change in all 3 dimensions. For example, flow around a
Note: Due to viscous effects (no-slip condition at the wall, Sec. 2.5 – 2.6) real flows are not truly 1D, but it is often sufficient to assume they are and the use of mean values in each plane is sufficient.
1.4. CONSERVATION LAWS In Fluid Mechanics, the vast majority of theories and methods are based on the following 3 conservation laws:
- conservation of mass - conservation of momentum - conservation of energy
These 3 principles will form the fundamentals of Computational Fluid Dynamics (CFD) too. We will derive them specifically for CFD purposes in the next Chapter, however, for now let us recall their meaning and the form in which we have learned them in Fluid Mechanics I. 1.4.1. Control Volume Recall that the conservation laws were applied to a Control Volume in Fluid Mechanics. Control Volume is a specific, suitably chosen volume, in which we monitor or “control” the behaviour of fluid. (Note the analogy to the Transit Zone at an Airport – we control who can get in and who gets out and as such, this is a “controlled zone” of people. A “controlled zone” for fluid particles – what we call a Control Volume - is the same: we monitor in it that what goes in and what goes out.) It can be large or small, and of any shape. In CFD, we will use thousands or millions of very small hexahedral (cube) or tetrahedral (pyramid) shape Control Volumes.
Control Volume of hexahedral shape Control Volume of tetrahedral shape
Advantage: we only need to know the properties at the edges of the Control Volume, not inside. This allows us to get around the complexities of the objects or machines inside the Control Volume. Example: if we put a Control Volume (CV) around a jet engine and we know what is the inflow and outflow of the CV, we can calculate the thrust of the engine even without considering the complexities of the jet engine itself.
NGM_JF006_1: Computational Fluid Dynamics Széchenyi University Instructor: D. Feszty, T. Jakubík Audi Hungaria Dept. of Whole Vehicle Development
1.4.2. Conservation of Mass This law essentially says: mass outflow – mass inflow = decrease of mass in CV net rate of mass in CV For a Control Volume as the one shown below, the conservation of mass will take the form (see Fluid Mechanics I for full derivation):
Control Surface (CS)
NGM_JF006_1: Computational Fluid Dynamics Széchenyi University Instructor: D. Feszty, T. Jakubík Audi Hungaria Dept. of Whole Vehicle Development
For uniform 1D flow (replace the integral with summation) :
… in other words (for steady, 1D flows), what goes in, must come out: fluid (or mass) cannot disappear!
1.4.3. Conservation of Momentum This law is based on Newton’s 2nd law:
�⃗� = 𝑚�⃗� = 𝑚𝑑�⃗⃗�
𝑑𝑡
and it allows to determine the force (F) which the fluid exerts on its surroundings. For the Control Volume shown earlier, it will take the form of (see Fluid Mechanics I for derivation):
fluid velocity relative to the CV surface, i.e. the inflow or outflow velocity from the Control Volume fluid velocity relative to an inertial reference frame. Makes a big difference for moving Control Volumes
NGM_JF006_1: Computational Fluid Dynamics Széchenyi University Instructor: D. Feszty, T. Jakubík Audi Hungaria Dept. of Whole Vehicle Development
1.5. SOLUTION ALONG A STREAMLINE We can derive useful solutions for fluid flows along a streamline too, i.e. without using the Control Volume method. Such solution allows us to assess the fluid properties between two points along a streamline.
The most widely used method for a solution along a streamline is the Bernoulli equation, which
was derived for the following assumptions:
- incompressible fluid (constant density, i.e. = const)
- inviscid fluid (zero viscosity, i.e. = 0)
- no energy added or extracted between the two points
- losses due to friction are negligible
NGM_JF006_1: Computational Fluid Dynamics Széchenyi University Instructor: D. Feszty, T. Jakubík Audi Hungaria Dept. of Whole Vehicle Development
static pressure (p) pressure, which one would feel when moving along with
the particles in the fluid
dynamic pressure pressure associated with the motion of the fluid
stagnation pressure (p0) pressure, which would occur when the fluid is brought to
rest. For incompressible fluid (note: this is NOT valid for
compressible fluid)
Bernoulli equation essentially says that for a fluid where elevation effects are negligible
(gz along a streamline is constant), the stagnation pressure (p0) along a streamline is
constant. Warning!
- It is a common mistake for students to use the Bernoulli equations for any case in Fluid Mechanics. This is wrong! Bernoulli equation can only be used for the conditions listed among the assumptions made above, i.e. incompressible, inviscid, 1D flow.
- It is a big mistake to apply it to compressible flows or to the flow across
machines, which generate or extract energy, such as wind turbines, propellers, pumps, jet engines, etc!!
With this, we have completed the review of the relevant fundamentals from Fluid Mechanics. When we apply the conservation of mass, momentum and energy to a small fluid element, then we obtain the governing equations of Computational Fluid Dynamics, the so-called Navier Stokes equations. These will be derived in the next Chapter.