The velocity vector, for an incompressible fluid, must satisfy the continuity equation

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1 Chapter D Vortex Methods Vortex methods are powerful numerical methods to study the evolution of inviscid flow fields. The idea behind the method will be developed in the following notes.. Vorticity and Streamfunction The 3D vorticity is defined as the curl of the velocity vector: ω = u (.) The velocity vector, for an incompressible fluid, must satisfy the continuity equation u =. (.) This condition is immediately satisfied by introducing a vector streamfunction Ψ such that the velocity is its curl: u = Ψ (.3) since vector identities guarantees that the divergence of a curl is always zero u = Ψ =. The vorticity can now be written in terms of the streamfunction guaranteeing that the resultant flow field is divergence-free: ω = Ψ = ( Ψ) Ψ (.4) The above equation can be further simplified by requiring that the streamfunction vector be divergence-free. We then obtain a vector Poisson equation for the streamfunction for a given vorticity field: Ψ = ω (.5) 99

2 CHAPTER. D VORTEX METHODS The situation is considerably simpler in D where the streamfunction is a scalar, ψ, and where only the vorticity component perpendicular to the plane is non-zero ω = ωk. The vector Poisson equation reduces to a scalar Poisson equation: ψ = ω (.6) The important point about vorticity is that for two-dimensional inviscid flows, and in the absence of vorticity sources, the equation governing the evolution of the vorticity is simple and states that the vorticity is conserved following the flow: dω dt = ω + u ω = (.7) t The idea behind vortex method is then simply to represent the vorticity fields as a collection of vortex patches carrying an intrinsic amount of vorticity. This vorticity has associated with it a flow field which in turn will advect these patches around without changing their intrinsic vorticity. The flow field will of course change once the patches change position, and one merely has to track them in order to simulate the flow evolution.. Flow due to point vortices Here we derive the relationship between a vorticity source and the induced streamfunction and velocity in an infinite plane with no boundaries. This derivation uses the Green function approach where the forcing, the vorticity source, is concentrated at points and has infinite strength in a pointwise sense, but finite strength in an area-averaged sense. We first look at the impact of a single point vortex then use the linearity of the equations to superpose solution when multiple vortex points are located... Flow due to single point vortex An analytical expression for the streamfunction equation can be derived for the special case where there is a single impulse vorticity in an infinite domain ψ = ω δ(r) πr, r = x x, r = x x (.8) where ω is now simply the strength of a concentrated vortex point. The Green function solution is then simply ψ(x) = ω(x ) ln r π = ω(x ) ln x x π (.9)

3 .. FLOW DUE TO POINT VORTICES The velocity associated with this streamfunction field is then: u = (ψk) = ω(x ) (rk) πr = ω(x ) r k πr (.) Since r = (x x ) + (y y ) we have r = r/r. The velocity induced by a point vortex with strength ω is then: u(x) = ω(x ) r k πr = ω(x ) (x x ) k π x x = ω(x ) k (x x ) π x x (.).. Flow due to multiple point vortices The linearity of equations.8,., and. makes it possible to invoke the principle of superposition to account for the presence of multiple point vortices in the flow. Tagging each point vortex by a subscript i, we can immediately write down the expression of the streamfunction and velocity imparted by each of them to the point located at x: ψ i (x) = ω(x i ) ln x x i π (.) u i (x) = ω(x i ) k (x x i) π x x i (.3) where x i is the location of the i-th vortex and ω(x i ) is its strength. The total imparted velocity/streamfunction can be obtained by summing the individual contributions: ψ(x) = ψ i (x) = u(x) = u i (x) = ω(x i ) ln x x i π (.4) ω(x i ) k (x x i) π x x i (.5) Figure. shows the streamfunction field and the velocity induced by,, 3 and 4 point vortices of similar strengths. The streamlines are shown by contours and the velocity field by arrows. If an infinite number of vortex element are present, the summation becomes an area integral: u(x) = ω(x ) k (x x ) dx Ω π x x (.6) The integration is carried out on the location of the impulse x, and where dx represents an area element. The above equation is often written in the form u(x) =

4 CHAPTER. D VORTEX METHODS Figure.: Streamfunction contours and flow field induced by a unit-strong vortex located on the diamond (±, ) and (, ±).

5 .3. FROM POINTS TO BLOBS 3 K(x x ) ω(x ) where the the stands for a convolution operation and K is called the Kernel. Equation.6 is the D version of the Biot-Savart law, and can be seen as the continuous form of equation.5. We have derived the continuous Biot-Savart law starting from discrete impulses to continuum forcing of the form ω(x ). It is useful to reverse this trajectory and re-derive the discrete system starting from the continuum. If the vorticity consists of particles carrying a certain amount of vorticity distributed according to: ω i (x ) = α i ζ i ( x x i ) with ζ( x x i ) as x x i (.7) where α i is the circulation around the i th particle. We require further that ζ decays fairly quickly with distance from the center. u(x) = α i Ω ζ(x x i ) k (x x ) π x x dx (.8) In the limiting case where the ζ(x x i ) is the D Dirac delta function the expression simplifies to: k (x x i ) u(x) = α i (.9) π x x i.3 From points to blobs One problem with approximating the vorticity with delta functions, i.e. infinite spikes, is the associated singularity in the streamfunction and velocity fields. This singularity appears when the distance to the point vortex becomes very small, r = x x = : both streamfunction and velocity become very large; the first blows up logarithmically fast and the other like r. This is quite problematic in the evaluation of the sums. It is therefore most useful to replace the spiky representation of the vorticity by blobs; these blobs will have finite widths and hence the associated singularity is removed. The questions become how do we construct the vorticity representation of the blobs, and what are the streamfunction and velocity Kernels associated with it? In order to build these representations we retrace the path followed in the derivation of the point vortex methods. We thus assume that each vortex blob has associated with it a localized shape function, ω = ζ(r) that is radially symmetric about the center of the blob, that decays as we move away from the center, and whose integral over the entire plane is unity. Under these assumptions, the solution to the Poisson equation is still expected to be radially symmetric and reduces to: r ( r r G r ) = ζ(r) (.)

6 4 CHAPTER. D VORTEX METHODS where ζ(r) is yet an undetermined function and G its corresponding Green function. The above equation can be integrated twice to yield: G r = r G(r) = r r t sζ(s) ds (.) ( t ) sζ(s) ds dt (.) Here r and s are integration variables. Once a vortex representation ζ(r) is chosen, equation. can be used to compute the velocity kernel, and equation. to compute the streamfunction kernel. It is the former that is most critical and we have K(x) = k (x x ) G r r = k (x x ) r sζ(s) ds (.3) r.3. Gaussian blobs of second order A common choice for the representation of the vorticity is to use a Gaussian function that has the form: /ǫ ζ(r) = e r (.4) πǫ where ǫ is the width of the blob, the proportionality factor is simply for normalization so that the integral of the above function over the infinite plane is. The vorticity distribution peaks at at r = and decays exponentially fast as we πǫ get away from the blob center; the parameter ǫ determines how fast the function decays. Notice that this function approaches the Dirac delta function as ǫ : its width narrows to a point, its amplitude goes to, and its area integral remains unity. The corresponding velocity kernel can be obtained by plugging the above expression in. and evaluating the integral to obtain: K(x) = k (x x ) πr ( e r /ǫ ) (.5) Figure. shows the impact of the width parameter on the vortex blob. The primary effect is to remove the effect of the singularity in the vicinity of the blob center while mimicking the behavior of a point vortex as we move away from the center..3. Gaussian blobs of high order The Gaussian blobs of the previous section can be shown to be second order in the parameter ǫ, it is however, possible to derive higher order shape functions: ζ 6 (r) = ǫ π ζ 4 (r) = ǫ ( ) 4 r e r π K 4 = k (x x [ ) + ( r ) e ] r π x x ( 6 6 r + r 4) e r K 6 = k (x x ) π x x [ ( r + ) ] r4 e r

7 .3. FROM POINTS TO BLOBS 5 Figure.: Left panels: vortex blob Gaussian decay for several values of the blob width ǫ:.4 (blue),. (green) and. (red). The right panels show the velocity Kernel amplitude as a function of r; the black line refers to the point vortex case. Top to bottom are the Gaussian blobs of order, 4 and 6, respectively.

8 6 CHAPTER. D VORTEX METHODS where r = r/ǫ is a scaled radius. The radial shape functions associated with the 4 and 6-th order Gaussian blobs are shown in figures.. Again, the trend is clear in that a Dirac delta function is approached as the width parameter ǫ shrinks, and as the order increases. The corresponding velocity kernel amplitudes are shown in the left panels. They all mimic the behavior of the point vortex away from the blob center but decay to zero once at the vortex center instead of behaving singularly..4 The vortex method algorithm In this section we summarize the steps required to implement the vortex method as a computational code. The steps are as follows: Choose a shape function for the vorticity and the associated velocity Kernel. To make the steps more concrete we assume that we have chosen the second order Gaussian distribution and its velocity Kernel. So we have the following approximations for the vorticity field and velocity: ω(x) = u(x) = e r i α i (.6) πǫ α i k (x x i ) π x x i ( e r ) (.7) r = x x i, r i = r i /ǫ (.8) Determine the width, ǫ, positions, x n i and strengths, α n i of the vortex blobs at an initial time; here the superscript n refers to the time step. Determine the velocity field induced by all vortex blobs on all other blobs: use equation.7 to compute u(x j ) for all particles j =,,..., N. Update the particles position by integrating the equation dx i dt = u(x j) (.9) using a Runge-Kutta 4 integrator (this requires 4 stages). In the absence of vorticity sources, the strength of each vortex blobs is preserved, so α n i remains constant Return to step.4 and repeat until final time. throughtout the calculation.

9 .5. THE PRESENCE OF BOUNDARIES 7.5 The presence of boundaries The vortex method is ideal for boundaries-free problem where the flow domain is infinite. Accounting for boundaries in vortex methods is complicated provided the formulation is modified. The procedure hinges on the Helmholtz decomposition of a flow field into its potential (and irrotational) part and its solenoidal (and rotational part). The solenoidal part can be evolved with the vortex method ignoring the presence of boundaries. The potential part is then introduced to correct the solenoidal fields and account for the no-flow conditions at the solid boundaries. The potential part is governed by a Laplace equation and can be solved efficiently with a Boundary Element Method. The flow components can then be solved with points and lines only..5. The Helmholtz decomposition It is well known that any flow field u can be decomposed into u = u s + u p (.3) where u s = and u p =. These two properties can be guaranteed if we introduce a potential function and a streamfunction such that u p = φ and u s = Ψ. (.3) Inserting these expressions in the Helmholtz decomposition we obtain the following pair of equations u = ( Ψ + φ) = Ψ = Ψ (.3) u = ( Ψ + φ) = φ = φ (.33) where we have assumed again, and without loss of generality, that the streamfunction is divergence-free (this is certainly the case in D). Given the divergence and rotation of the original flow velocity equations.3 and.33 can be inverted for the streamfunction and potential, respectively. The solution steps are as follow:. The rotational part of the flow is evolved according to the vorticity evolution equation and the associated solenoidal velocity is obtained from the Biot- Savart law as outlined in the discussion on the vortex method. This velocity field will violate the boundary conditions of no-through flow.. The introduction of the potential velocity allows us to rectify the situation and enforce the no-through condition. For an incompressible flow the governing equation for the potential is the Laplace equation coupled with a Neumann boundary conditions at the boundaries: where Γ refers to the solid boundaries. φ =, and φ n = u s n on Γ (.34)