Two-Dimensional Potential Flow Session delivered by: Prof. M. D. Deshpande 1
Session Objectives -- At the end of this session the delegate would have understood PEMP The potential theory and its application i to 2-D Dirrotational i flows The flow field in simple flows like uniform, source, vortex, doublet flows The principle of superposition and its application to simple cases The method of images Flow around a cylinder and forces on a general 2-D cylinder Stokes Theorem, Kutta-Joukowski theorem Kutta condition 2
Session Topics 1. Basic Potentail Theory 2. 2-D Potential Flows 3. Simple Flows: Uniform, Source, Vortex, Doublet 4. Superposition of Flows 5. Method of Images 6. Flow around a Cylinder 7. Force on a General 2-D Cylinder 8. Circulation, Vorticity, Stokes Theorem 9. Kutta-Joukowski Theorem 10. Kutta Condition 3
2-D Potential Flow This chapter starts the description of solution methods in detail. Beginning with the simplest flows: two-dimensional, inviscid and irrotational, the chapter describes the basic theoretical results. These are applied to airfoil il problems in later chapters and then modified to include the effects of compressibility and viscosity. Basic Theory Sources and Vortices Interactive Calculations References 4
Basic 2-D Potential Theory PEMP We outline here the way in which the "known" solutions used in panel methods can be generated and obtain some useful solutions to some fundamental fluid flow problems. Often the known solutions can be applied, but sometimes other approaches are possible. The simplest case, two-dimensional potential flow illustrates this process. We shall discuss 2-D incompressible potential flow and just mention the extension to linearized compressible flow. For this case the relevant equation is Laplace's equation: 2 2 φ φ 2 Φ = 0, OR in 2-D + = 0 2 2. x y 5
There are several ways of generating fundamental solutions to this linear, homogeneous, second order differential equation with constant coefficients. Two methods are particularly useful: Separation of variables and the use of complex variables. Complex variables are especially useful in solving Laplace's equation because of the following: We know, from the theory of complex variables, that in a region where a function of the complex variable z = x + iy is analytic, the derivative with respect to z is the same in any direction. This leads to the famous Cauchy-Riemann conditions for an analytic function in the complex plane. 6
Consider the complex function: W = φ +iψ ψ The Cauchy-Riemann conditions are: Differentiating the first equation with respect to x and the second with respect to y and adding gives: Thus, analytic function of a complex variable is a solution to Laplace's equation and may be used as part of a more general solution. 7
W=φ+iψ ψ is called the complex potential. It consists of the usual velocity potential as the real part and the stream function as its imaginary part. The flow velocities can then be written as a single complex number: dw/dz = u - iv (Try deriving this.) We consider some simple analytic functions for W that are of great use in applied aerodynamics: Uniform flow W(z) = U z Line Source or Vortex W(z) = [K / (2 π )] ln z Doublet W(z) = C / (2 π z) 8
Uniform Flow If U is real the flow is in the x direction with a speed U. The flow direction can be adjusted by changing real and imaginary parts. This is a good example of the fact that the potential is not defined apart from an arbitrary constant. Although the flow is uniform everywhere, the potential depends on our choice of the origin. Differences in the potential are physically meaningful, though and do not depend on the choice of the origin. 9
Line Source or Vortex PEMP The same expression describes a "point" source or vortex in 2-D (which can be thought of as a vortex line or line of sources in 3-D). When K is real the expression describes a source with radially directed induced velocity vectors; imaginary values lead to vortex flows with induced velocities in the tangential direction. Further discussion of these flows is given in the next section. 10
Line Source or Vortex (Contd) Φ = (S / 2 π)lnr R Φ = (- Γ / 2 π) ln R 11
Doublet PEMP A doublet is formed by superimposing a source and a sink along the x-axis. The doublet strength is given by (S 2x 0 ). The fundamental doublet singularity with the potential shown above is formed by taking the limit as 2x 0 goes to zero and S goes to infinity while keeping the product constant. The doublet is commonly used as one of the fundamental singularities in many panel methods. Source-sink pair 12
Doublet (Contd) Streamlines of a source-sink pair and Streamlines of a doublet 13
Sources and dvortices 14
Notice that many of the solutions to the 2-D potential equation that we proposed are singular. In fact, the source solution seems the ultimate way of violating continuity while the vortex is the essence of rotational (not irrotational as we assumed) flow. These solutions are indeed singular at a point and do not satisfy the differential equation at that point. Away from the singularity, however, they are perfectly adequate solutions as can be seen by evaluating the integral forms of the continuity and irrotationality conditions. 15
Why the flow field near a source satisfies continuity: 16
Why the flow field near a vortex satisfies irrotationality: 17
The solutions are singular at a point, but even near the singular point strange things happen: the velocity gets very large. In real life, the large velocities in this region give rise to compressibility effects; viscous effects smear the discrete vortex into a distribution of vorticity in a viscous core. The actual velocity distribution near the core of a free vortex behaves more like a solid body with a velocity distribution V(R) = kr. (This is the result obtained by assuming a Gaussian distribution of distributed vorticity in the core region. The size of the viscous core depends on the Reynolds number, often taken as Γ/ν.). 18
Decay of a vortex filament in a viscous fluid. At t = 0, u θ = Γ /( 2 π r). Dashed lines correspond to the case of rigid body rotation corresponding roughly to core radii proportional to sqrt(ν t). From: Kuethe & Chow PEMP 19
This 1/r behavior of the vortex induced velocity is not just a mathematical result. It is essential for the flow to exist in equilibrium. We can easily see that the velocity must vary as 1/r for the pressure gradients to balance the centrifugal force acting on the fluid. The derivation is shown later. 20
Examples We can combine these singularities in different locations to produce the desired flow pattern. Since the solution to Laplace's equation is uniquely determined in regions without singularities when the solution on the boundaries is specified, we can use combinations of singularities to model many flows of interest. Ground Effects Cylinder Group of Vortices (Method of Images) (Source Doublets) (Stokes Theorem) 21
Streamlines for ψ = uniform flow (ψ 1 ) + Source (ψ 2 ) for V = 1 and h = Λ /2 V = L. Note, for instance, that the ψ = 0 streamline (OA & BAB ) is the locus of the intersections of the streamlines ψ 2 and ψ 1 ( =-ψψ 2 ). Similarly, for the ψ = 0.1 streamline, ψ 1 + ψ 2 = 0.1, and so forth. (From: Kuethe and Chow) PEMP NOTE: ψ 1 = V y Λ = Source strength ψ 2 = (Λ /2 π) θ ψ = ψ 1 + ψ 2 = V [(h θ / π ) - y] h = Λ /2 V = L =Characteristic length of the combined flow. Vel = 0, (h / π) ahead of the source location. 22
Flow pattern from distributing m identical line sources along the dashed line in the presence of a uniform flow. Only the upper half of the flow is shown. Circles indicate the locations of the line sources. (a) m = 5. (b) m = 11 (same total source strength). (c ) m = 101 (same total source strength). (d) m = 101 (but the source strength is reduced). (e) m infinity with λ /2 = V λ = Source density = flow rate /area. See that forward velocity is zero. Rear side velocity is λ /2 = V and normal to the panel. (f) Boundary condition at inclined panel. (From: Kuethe and Chow) 23
Derivation of Vortex Velocity Distribution ib ti The inward force on an element of fluid due to pressure gradients may be found by summing the contributions from the inner, outer, and side faces. 24
The result is: F = r dθ dp + higher order terms. Centrifugal force (outward): F = ρ r dr dθ V 2 / r For equilibrium: r dθ dp = ρ rdrdθ dθ V 2 /r So, dp = ρ V 2 dr / r, and from the Euler equation, dp = -ρ V dv. Integrating: g ρ V 2 dr / r = -ρ ρ V dv yields: V = constant / r 25
Mthd Method of fimages The flow field created by singularities in the presence of solid boundaries can be simulated by superimposing "image vortices". This works because the symmetry of the problem on the right ensures that there is no flow through the plane of symmetry. The boundary does the same thing for the problem on the left. Since both of these problems have the same boundary conditions and satisfy the same linear differential equation, the flow must be the same. 26
A source near a plane wall (From: Kuethe and Chow) 27
A vortex near a plane wall (From: Kuethe and Chow) 28
This technique is useful for simulating the effects of the ground on the aerodynamics of cars or airplanes at low altitude. It can also be used in more complex situations. Here, three images are required to simulate the boundary conditions associated with a corner. 29
This technique is used to predict the effects of wind tunnel walls on the flow field of models being tested. Imagine the system of image vortices that would be required to simulate wall effects on a 2D airfoil test. Yes, more than 2 images are required. The 3-D situation cannot in general be solved with images. 30
Motion of a vortex pair near the ground (From: Kuethe and Chow) 31
Cylinders Streamlines for (a doublet + Uniform flow): Synthesis of flow around a circular cylinder in uniform flow PEMP 32
Cylinders (Contd) PEMP The flow around a circular cylinder may be computed from a uniform stream and a doublet. (See previous sections.) The potential of the combined flow is: Differentiating to find the velocities gives: NOTE: On the cylinder wall, only radial velocity is zero. 33
Some interesting conclusions and generalizations follow from the expressions for the velocity and the potential on a circular cylinder shown above. Note that on the surface of the cylinder, the tangential velocity is: V = 2U sin θ,, so the maximum velocity is twice the freestream value. 34
Continuous distribution of doublets in a uniform flow (From: Kuethe and Chow) 35
The more general forms of these results hold for all ellipsoids: V max = U (1 + t/c) and V at surface = - n (n V max ) Notice that this holds exactly in incompressible potential flow, even if the ellipse has a t/c much larger than 1. Of course, in such a case, the real flow will probably look quite different from the potential flow solution. 36
The force on a general 2-D cylinder The force on a general 2-D cylinder can be computed by calculating the velocities, using Bernoulli's law to compute pressures, then integrating the surface pressures. However, the total forces and moments can be derived directly from the complex potential. The result is called the Blasius theorem. It is not derived d here, but the result follows from the theory of residues, the complex potential, and the incompressible Bernoulli equation. (Or one might just use the momentum equation and compute the net force by far field integrals.). 37
The force on a general 2-D cylinder (Contd) where Γ is the total circulation (measured counter-clockwise) and dsi is the net source strength. th In the case of no net source strength, the net force exerted on a collection of sources and vortices in a flow with freestream velocity U is perpendicular to the freestream and proportional to U and the total circulation. 38
Circulation, Vorticity, it and Stokes Theorem Stokes' theorem is an integral identity that may be written: The first integration is done over volume V. (dv missing). When the vector function F is taken to be the velocity field, V, then this relation in 2-D may be restated as: 39
This result implies that the circulation around a contour that contains a group of vortices is just equal to the sum of the enclosed vortex strengths. This allows application of the Blasius theorem to find the force acting on a group of vortices. The result is sometimes called the Kutta-Joukowski law: 40
Kutta-Joukowski Theorem As seen before this theorem states; L = ρ V Γ Note the direction of the lift force carefully. We can also treat the flow field far from a group of vortices as if it were created by a single vortex with a strength equal to the sum of the individual vortices. Such far field solutions can be especially simple and useful as a check of more complex results. Far field solutions can also be used as boundary conditions for the more complex near field solution, reducing the required extent of computational grids. 41
We should note here that just because we find a superposition of singularities that satisfies the boundary conditions and the differential equation, it does not mean that we have found the only solution to the problem. For example, we could add a vortex to the doublet that was used to model the circular cylinder, and we would still find that the flow went around the cylinder. These non-unique solutions are problemsome and we appeal to additional considerations to find the one(s) that actually will appear in nature. Just such an auxiliary condition, the Kutta condition, is provided by viscous effects which then determine the value of circulation. 42
Kutta condition We are considering steady, incompressible, irrotational 2-D motion. Kutta-Joukowski theorem states that force experienced by a body in a uniform stream acts perpendicular to the flow direction and is given by F = ρ V Γ. With the given boundary conditions on the boundary (normal vel = 0) and at infinity (uniform flow) we try to solve the Laplace equation. The flow is not unique. One way to make it unique is to specify the circulation. See that it is equivalent to specifying Lift itself! Then the formula above is not of practical use. In the figure below Γ is specified to be zero and hence we get zero lift. PEMP 43
Kutta condition (Contd) In the problem we are considering i if the circulation is specified the problem has a unique solution. The bodies we are considering are airfoils with a sharp trailing edge. They have the rear stagnation point at the trailing edge since the flow cannot take a sharp turn as shown in the last figure due to viscosity. Hence in real fluids the body with a sharp trailing edge will create enough circulation to hold the stagnation point at the trailing edge as shown in the figure below. This fixes the entire flow and the values of circulation and lift. Hence we are able to evaluate the value of lift by ideal fluid flow itself. PEMP 44
Free Vortices Singularities that are free to move in the flow do not behave in response to F = m a (what is m?). Rather they move with the local flow velocity. Thus, vortices and sources are convected downstream with the flow. And interacting singularities can produce complex motions due to their mutual induced velocities. 45
A pair of counter-rotating vortices moves downward because of their mutual induced velocities. Co-rotating vortices orbit each other under the influence of their mutual induced velocities. 46
Streamlines Past Sources and Vortices (Interactive Program) Drag any of the singularities from the well on the right into the main computation area. Set the freestream speed (the flow is from left to right), then click Compute. The marks on the page simulate small tufts and indicate the direction of the local flow. Experiment with multiple singularities to simulate a pair of wing trailing vortices, a source/sink doublet, or a spinning baseball. If you do not see the results you may want to try an alternate version of this applet. Due to certain platform-dependent java problems, this program may not work with some browsers on some platforms. If not, try here for a less cool, but simpler version of the program. 47
Summary The following topics were dealt in this session The Potential theory and its application to 2-D irrotational flows The flow field in simple flows like uniform, source, vortex, doublet flows Application i of superposition i principle i to simple flows Method of images Flow around a cylinder and the forces involved Stokes theorem, Kutta-Joukowski theorem Kutta condition 48
Thank you 49