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530 chapter 1919.8 2-D Flow over a BeamNow that the comparison with an analytic solution has shown that our CFD simulationworks, we return to determining if the beam in Figure 19.4 might producea good resting place for salmon. While we have no analytic solution with which tocompare, our canoeing and fishing adventures have taught us that standing waveswith fish in them are often formed behind rocks in streams, and so we expect therewill be a standing wave formed behind the beam.19.9 Theory: Vorticity Form of theNavier–Stokes EquationWe have seen how to solve numerically the hydrodynamics equations⃗∇·v =0, (v · ⃗∇)v = − 1 ρ ⃗ ∇P + ν∇ 2 v. (19.52)These equations determine the components of a fluid’s velocity, pressure, and densityas functions of position. In analogy to electrostatics, where one usually solvesfor the simpler scalar potential and then takes its gradient to determine the morecomplicated vector field, we now recast the hydrodynamic equations into formsthat permit us to solve two simpler equations for simpler functions, from whichthe velocity is obtained via a gradient operation. 3We introduce the stream function u(x) from which the velocity is determined bythe curl operator:(v def= ∇×u(x)=ˆɛ ⃗ ∂uzx∂y − ∂u ) (y ∂ux+ˆɛ y∂z ∂z − ∂u )z. (19.53)∂xNote the absence of the z component of velocity v for our problem. Since ⃗ ∇·( ⃗ ∇×u) ≡ 0, we see that any v that can be written as the curl of u automatically satisfiesthe continuity equation ⃗ ∇·v =0. Further, since v for our problem has only x andy components, u(x) needs to have only a z component:u z ≡ u ⇒ v x = ∂u∂y ,v y = − ∂u∂x . (19.54)(Even though the vorticity has just one component, it is a pseudoscalar and not ascalar because it reverses sign upon reflection.) It is worth noting that in 2-D flows,the contour lines u = constant are the streamlines.3 If we had to solve only the simpler problem of irrotational flow (no turbulence), then wewould be able to use a scalar velocity potential, in close analogy to electrostatics [Lamb 93].For the more general rotational flow, two vector potentials are required.−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 530