INVISID ,IMCOMPRESSIBLE FLOW
Ch 3 Anderson
Jan 04, 2016
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INVISID ,IMCOMPRESSIBLE FLOW
IMPORTANCE
From an aerodynamic point of view, at air velocities between 0 and 300 mi/h the air densityremains essentially constant, varying by only a
few percent.
IMPORTANCE
l most modern general aviation aircraft still fly at speeds below 300 mi/h
l , the principles of incompressible flow apply to the flow of fluids, e.g., water flow through pipes, the motion of submarines and ships through the ocean, the design of wind turbines (the modern term for windmills), and many other important applications.
BERNOULLI'S EQUATION
Condition
l Incompressible flow
p +ρ/2*V^2 +ρ*g*z = const
that Bernoulli's equation was derived from the momentum equation;hence, it is a statement of Newton's second law for an inviscid,
incompressible flow with no body forces.Bernoulli's equation can be derived from the general energy equation.
Rotational vs Irrotational
l For a general, rotational flow, the value of the constant in BERNOULLI'S EQUATION will change from one streamline to the next so it's applied only from the same streamline.
if the flow is irrotational, then Bernoulli's equation holds between any two points in the flow, not necessarily just on the same streamline.
INCOMPRESSIBLE FLOW IN A DUCT:THE VENTURI AND LOW-SPEED
WIND TUNNEL
Quasi-one-dimensional flow
l A = A(x), V = V(x), p = p(x),
l quasi-one-dimensional continuity equation:ρ1*A1*v1=ρ2*A2*v2
In physical terms, it states that the mass flow (Kg per second) through the duct is constant.
For incompressible flowA1*v1=A2*v2
In physical terms, it states that the volume flow (cubic feet per second or cubic meters per
second) through the duct is constant. We see that if the area decreases along the flow
(convergent duct), the velocity increases then pressure decreases from Bernoulli's equation;
conversely, if the area increases (divergent duct), the velocity decreases then pressure increase.
venturi
l a venturi can be used to measure airspeeds. Consider a venturi with a given inlet-to-throat area ratio A1/ A 2 , Assume that the venturi is inserted into an air stream that has an unknown velocity V1 We wish to use the venturi to measure this velocity. With regard to the venturi itself, the most direct quantity that can be measured is the pressure difference PI – P2. This can be accomplished by placing a small hole (a pressure tap) in the wall of the venturi at both the inlet and the throat and connecting the pressure leads (tubes) from these holes across a differential pressure gage, or to both sides of a U-tube manometer. In such a fashion, the pressure difference PI - P2 can be obtained directly. This measured
l pressure difference can be related to the unknown velocity V1
V1^2=2/ρ*(p2-p1)+V2^2V2=A1*V1/A2
l a low-speed wind tunnel is a large venturi where the airflow is driven by a fan connected to some type of motor drive. The wind-tunnel fan blades are similar to airplane propellers and are designed to draw the airflow through the tunnel circuit.
l The wind tunnel may be open circuit, where the air is drawn in the front directly from the atmosphere and exhausted out the back, again directly to the atmosphere.
l the wind tunnel may be closed circuit, where the air from the exhaust is returned directly to the front of the tunnel via a closed duct forming a loop.
l The basic factor that controls the air velocity in the test section of a given low-speed wind tunnel is the pressure difference PI – P2 as The area ratio A2/A1 is a fixed quantity for a wind tunnel of given design. Moreover, the density is a known constant for incompressible flow.
V1^2=2/ρ*(p2-p1)+V2^2V2=A1*V1/A2
l Static pressure
l Static pressure is the pressure you feel by moving with the flow at its local velocity VI
l Measured by making The plane of the hole of manometer is parallel to the flow.
l
Stagnation pressure
l Also total pressure and it's the pressure that the flow achieves when the velocity is reduced to zero.
l Measured by making The plane of the hole of manometer is perpendicular to the flow.
Flow speed calculation
l In the case for incompressible flow, using pitot tube and applying in this law
l v=(2*(Po-P)/ρ)^.5
dynamic pressure
l = .5*ρ*v^2 l Dynamic pressure is the kinetic energy per unit volume of a fluid particle.
l for incompressible flow, the dynamic pressure has special meaning; it's precisely the difference between total and static pressure.
PRESSURE COEFFICIENT
l Cp=(p-p
a
)/qa
l Qa
=.5*ρa*v
a2
l For incompressible flow from bernoullil C
p=1-(v/v
a)2
l the pressure coefficient at a stagnation point in an incompressible flow is always equal to 1.0.
This is the highest allowable value of Cp anywhere in the flow field.
PRESSURE COEFFICIENT
l For compressible flows, Cp at a stagnation point
is greater than 1.0l in regions of the flow where V>V
a or P< P
a, C
p
will be a negative value.
PRESSURE COEFFICIENT
l Cp should depend only on the Mach number,
Reynolds number, shape and orientation of the body, and location on the body.
l For incompressible flow, Cp is a function only of
location on the surface of the body, and the body shape and orientation.
Laplace's equation
condition
▼.v=0 l that's incompressible law
l Take care irrotational flow isl ▼*v=0
l But he continue as Laplace is for irrotational
GOVERNING EQUATION FORIRROTATIONAL, INCOMPRESSIBLE FLOW:LAPLACE'S EQUATION
LAPLACE'S EQUATION forms
l ▼2Ø=0
l For 2 dimension
From equations:l 1. Any irrotational, incompressible flow has a velocity potential and stream function (for two-dimensional flow) that both satisfy Laplace's equation.
l 2. Conversely, any solution of Laplace's equation represents the velocity potential or stream function (two-dimensional) for an irrotational, incompressible flow.
l Note that Laplace's equation is a second-order linear partial differential equation.
l It's Linear because the sum of any particular solutions of a linear differential equation is also a solution of the equation.
l Ø=Ø1+Ø2+Ø3+.......Øn
Boundary Conditions
l 1) infinite boundary conditions
l 2) Wall Boundary Conditions:l for inviscid flows the velocity at the surface can be finite, but because the flow cannot penetrate the surface, the velocity vector must be tangent to the surface. then the component of velocity normal to the surface must be zero. Let n be a unit vector normal to the surface
l If we are dealing with stream function rather than potential flow the boundary equation will be
where s is the distance measured along the body surface.l Note that the body contour is a
streamline of the flow. Recall that ψ = constant is the equation of a streamline and it is an alternative expression for the boundary
condition
l If we are dealing with the velocity components u and v themselves
INTERIM SUMMARY
So on we deal with two dimensional steady flows
UNIFORM FLOW
UNIFORM FLOW
l The flow is defined as uniform flow when in the flow field the velocity and other hydrodynamic
parameters do not change from point to point at any instant of time
l بسممكن الزمن فى دالة هو الفلويد من النوع فهذاهنا فتكون مقطع اى عند ثابتة الخواص انه قاصدة يكون
عشان واحد اتجاه فى وبيكون االفقى المحور فى دالةirrotational كده
l in general it's function for steady flowl Ø=V
a*x for com and incom
l Ψ=Va*y= constant(so flow in x direction) for
incom onlyl
SOURCE FLOW
l Consider a two-dimensional, incompressible flow where all the streamlines are straight lines emanating from a central point. Moreover, let the velocity along each of the streamlines vary inversely with distance from point 0. Such a flow is called a source flow.
SOURCE FLOW
Vθ = 0 , V
r=C/r
l where c is a constantl May be com or incoml Irrotational except at origin it's infinity
l In a source flow, the streamlines are directed away from the origin
l In a sink flow, the streamlines are directed toward the origin
l consider a depth of length I perpendicular to the page, a length I along the z axis. there is an entire line of sources along the z axis.
mass flow across the surface element dSl =2*π*r*l*ρ*V
r
l volume flow=2*π*r*l*Vrl A = volume flow / l = 2*π*r*Vr
l so Vr=A/(2*π*r)
l so c=A/(2*π)
l the source strength (A) : it is physically the rate of volume flow from the source, per unit depth perpendicular to the page (m2/s)
COMBINATION OF A UNIFORMFLOW WITH A SOURCE AND SINK
l Consider a polar coordinate system with a source of strength (A) located at the origin. Superimpose on this flow a uniform stream with velocity V
a moving from left to right.
l The stream function for the resulting flow is the sum which is Laplace's function as it a sum of two Laplace's equations
l Velocity:
l The stagnation points located at (r,θ) =(A/2πV
a, π) from
l Streamline equation which go through stagnation point
l Ψ = A/2
l This streamline is shown as curve ABCl All the fluid outside ABC is from the freestream, and all the fluid inside ABC is from the source.
Since we are dealing with inviscid flow, We can replace the solid surface by the source which surface have the same equation of Ψ = A/2 which can used to describe a semi-infinite body
l To close the body but a sink after point D (source) let it be here have the strength of the source (equal ,and opposite)
l The stream function=
l The 2 stagnation points are located as
l the stagnation streamline equation :
l Ψ =0
l I suppose if we the change the strength and position of the sink the body surface will change and we can reach to unsymmetric airfoil
How pressure can't affect source, sink ,and uniform flow
DOUBLET FLOW
This is a special, degenerate case of a source-sink (equal but opposite) separated by a distance I (l approach zero as I*A remains constant(???
l =0 ,and A approaches infinty))
l Again the stream function of source is
l The stream function for a doublet
l The strength of the doublet is denoted by K and is defined as k= I*A.
l velocity potential for a doublet
l Where r = d sin (θ) ,and d is diameter of a circle with on the vertical axis between source and sink and with the center located d / 2 directly above the origin.
l So we see we see that the streamlines for a doublet are a family of circles with diameter K/2πC.
l The different circles correspond to different values of the parameter c.
l In the previous figure (slide 59): the direction of flow is out of the origin to the left and back into the origin from the right.
l we designate the direction of the doublet by an arrow drawn from the sink to the source as in the previous figure
l If the arrow(direction) reversed, the sign reversed
NONLIFTING FLOW OVER ACIRCULAR CYLINDER
( the combination of a uniform flow and a doublet produces the flow over a circular cylinder)
l Consider the addition of a uniform flow with velocity V
a, and a doublet of strength K, The
direction of the doublet is upstream, facing into the uniform flow.
l the stream function for the flow over a circular cylinder of radius R or combination of free stream, and droplet.
l Velocity will be
l For this combination, there are two stagnation points, located at (r, θ) = (R, 0) and (R,π).
l The same streamline goes through both stagnation points, Moreover, the equation of this streamline: Ψ =0, and radius of the cylinder: R2 = K /2πV
a
l Satisfying stream function in by θ= π and θ = 0 for all values of r we find that Ψ =0; hence, the entire horizontal axis through points A and B, extending infinitely far upstream and downstream, is part of the stagnation streamline.
l the entire flow field is symmetrical about both the horizontal and vertical axes through the center of the cylinder; Hence, the pressure distribution is also symmetrical about both axes.
l the pressure distribution is symmetrical about both axes, so there is no net lift nor net drag (inviscid so no wake).
l The velocity distribution on the surface of the cylinder
l At the bottom , the angle is negative as sin(θ) is negative
l Vθ is negative because of the direction of flow in
the direction of decreasing angle at the top of cylinder
l Maximum velocity at the cylinder will be 2Va
l θ=π/2 , θ=3π/2
l the surface pressure coefficient over a circular cylinder is
l Cp varies from 1.0 at the stagnation points to -
3.0 at the points of maximum velocity.
Pressure distribution
VORTEX FLOW
l Consider a flow where all the streamlines are concentric circles about a given point. Moreover, let the velocity along any given circular streamline be constant, but let it vary from one streamline to another inversely with distance from the common center.
l vortex flow is a physically possible incompressible flow, and Vortex flow is irrotational everywhere except at the point r = 0, where the vorticity is infinite
l If we take the circulation around any curve not enclosing the origin, the result that Γ = o. It is only when we choose a curve that encloses the origin, where vorticity is infinite
l Vr=0
Vortex velocities
l Γ is called the strength of the vortex flow, and the circulation taken about all streamlines is the same value, namely, Γ = -2πC
l Vθ is negative when Γ is positive; a vortex of
positive strength rotates in the clockwise direction.
l The velocity potential
l The stream function
LIFTING FLOW OVER A CYLINDER SPINNING(Magnus effect)
l Consider the flow synthesized by the addition of the nonlifting flow over a cylinder and a vortex of strength r,
l stream function
l if r = R, then Ψ = 0 ,and is a valid stream function for the inviscid, incompressible flow over a circular cylinder of radius R
l no longer symmetrical about the horizontal axis, so there is finite lifting force
l There is symmetrical about the vertical axis, so there is no drag force (inviscid).
l Velocity :
l stagnation points
l The equation gives two stagnation points on the bottom half of the circular cylinder in the third and fourth quadrants.
l If Γ/4πVaR < 1 so there is 2 stagnation points
l If Γ/4πVaR = 1 so there is only one stagnation
point (R, -π/2)l If π so the previous equation has no meaning
l For Γ/4πVaR = 1 there are two stagnation points,
one inside and the other outside the cylinder, and both on the vertical axis, as shown by points 4 and 5 in Fig
l Kutta-loukowski theorem ( the lift per unit span)
What creating lift in spinning cylinder
l the friction between the fluid and the surface of the cylinder tends to drag the fluid near the surface in the same direction as the rotational motion. Superimposed on top of the usual nonspinning flow, this "extra" velocity contribution creates a higher-than-usual velocity at the top of the cylinder and a lower-than-usual velocity at the bottom, These velocities are assumed to be just outside the viscous boundary layer on the surface. from Bernoulli's equation that as the velocity increases, the pressure decreases. the pressure on the top of the cylinder is lower than on the bottom, This pressure imbalance creates a net upward force (a finite lift).
nonspinning cylinder
Spinning cylinder: peripheral(spinning) velocity of the
surface = 3 V( one stagnation point)
Spinning cylinder: peripheral velocity of the surface = 6 V (separated stagnation point)
l It is interesting to note that a rapidly spinning cylinder can produce a much higher lift than an airplane wing of the same planform area; however, the drag on the cylinder is also much higher than a well-designed wing.
l There are 4 points on the spinning where the P=P
a 2 on the top ,and 2 on the bottom
l For lift coefficient = 5 the pressure distribution is
THE KUTTA-JOUKOWSKI THEOREMAND THE GENERATION OF LIFT
Kutta-loukowski theorem ( the lift per unit span)l valid to cylindrical bodies of arbitrary cross section. for example airfoils.
As in ch.4
l As was said for vortex and spinning cylinder , the flow outside the airfoil is irrotational, and the circulation around any closed curve not enclosing the airfoil.
l the flow over an airfoil is synthesized by distributing vortices either on the surface or inside the airfoil.
l The Kutta-Joukowski theorem is simply an alternate way of expressing the consequences of the surface pressure distribution
NONLIFTING FLOWS OVERARBITRARY BODIES: THE NUMERICAL
SOURCE PANEL METHOD
l Here we specify the shape of an arbitrary body and solve for the distribution of singularities which, in combination with a uniform stream, produce the flow over the given body.
l This section is limited for the present to nonlifting flows.
The extension the concept of a source or sink
l imagine that we have an infinite number of such line sources side by side, where the strength of each line source is infinitesimally small. These side-by-side line sources form a source sheet, the "source" sheet is really a combination of line sources and line sinks.
l Let s be the distance measured along the source sheet in the edge view. Define λ = λ (s) to be the source strength per unit length along s (in the z direction) and per unit length (in the s direction).
l Typical units for λ are meters per second
l consider point P in the flow, located a distance r from ds.
l velocity potential at point P from source sheet
l Let us approximate the source sheet by a series of straight panels. Moreover, let the source strength A per unit length be constant over a given panel, but allow it to vary from one panel to the next. These panel strengths are unknown; the main thrust of the panel technique is to solve for λ
j , j = 1 to n,
l This boundary condition is imposed numerically by defining the midpoint of each panel to be a control point and by determining the λ
j's such
that the normal component of the flow velocity is zero at each control point.
l the potential velocity at point P(as in the fig) due to all the panels.
As rpj be the distance from any point on
the jth panel to P
l Since point P is just an arbitrary point in the flow, let us put P at the control point of the ith panel. Let the coordinates of this control point be given by (X
i, y
i)
l The normal component of velocity induced at (x j
, yJ) by the source panels is
l the derivative is simply λi/2 for j = i . so
the summation is the normal velocity induced at the ith control point by all the other panels.
l The normal component of the flow velocity at the ith control point is the sum of that due to the freestream and that due to the source panels. The boundary condition states that this sum must be zero.
Equation is the crux of the source panel method. The values of the integrals in depend simply on the panel geometry. they are not properties of the flow. Equation is a linear algebraic equation with n unknowns λI, λ 2 ,..., λn.
l ن عندنا فيبقى المعادلة فى نقطة كل نعوضعنمعادلة
l This approximation can be made more accurate by increasing the number of panels.
l For an airfoil, it is desirable to cover the leading-edge region with a number of small panels to represent accurately the rapid surface curvature and to use larger panels over the relatively flat portions of the body.
l Let S be the distance along the body surface, measured positive from front to rear.
l The total surface velocity at the ith control point is the sum of the contribution from the freestream, and from the source panels
The tangential velocity on a flat source panel induced by the panel itself is zero. because the panel can only emit volume flow from its surface in a direction perpendicular to the panel itself.
Answer check
l the strength of the jth panel itself is λjS
j. For a
closed body, the sum of all the source and sink strengths must be zero, or else the body itself would be adding or absorbing mass from the flow
THE FLOW OVER A CIRCULARCYLINDER-THE REAL CASE
( a viscous, incompressible flow)
l the drag coefficient is a function of the Reynolds number.
l 1/ For very low values of Re, say, 0 < Re < 4, the streamlines are almost symmetrical, and the flow is attached. This regime of viscous flow is called Stokes flow.
l 2/ For 4 < Re < 40, the flow becomes separated on the back of the cylinder, forming two distinct, stable vortices that remain in the position
l 3/ As Re is increased above 40, the flow behind the cylinder becomes unstable the vortices which were in a fixed position now are alternately shed from the body in a regular fashion and flow downstream. The alternately shed vortex pattern is called a Karman vortex street, named after Theodore von Karman
l 4/ As the Reynolds number is increased to large numbers, the Karman vortex street becomes turbulent and begins to metamorphose into a distinct wake. The laminar boundary layer on the cylinder separates from the surface on the forward face, at a point about 80° from the stagnation point. The value of the ,Reynolds number for this flow is on the order of 105 .
l 5/ For 3x105 < Re <3 x 106 , the separation of the laminar boundary layer still takes place on the forward face of the cylinder. However, in the free shear layer over the top of the separated region, transition to turbulent flow takes place. The flow then reattaches on the back face of the cylinder, but separates again at about 120° around the body measured from the stagnation point.
l This transition to turbulent flow, and the correspondingly thinner wake reduces the pressure drag on the cylinder and is responsible for the precipitous drop in C
D
l 6/ For Re < 3 *106 , the boundary layer transists directly to turbulent flow at some point on the forward face, and the boundary layer remains totally attached over the surface until it separates at an angular location slightly less than 120° on the back surface. For this regime of flow, CD actually increases slightly with increasing Re because the separation points on the back surface begin to move closer to the top and bottom of the cylinder, producing a fatter wake, and hence larger pressure drag.
real pressure distributions ,and The theoretical pressuredistribution over the surface of a cylinder in an inviscid,
incompressible
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