4. What ifRe→ ∞? (Excludey=hfrom this analysis).
Question #3: 2D Taylor-Green VortexTaylor–Green vortex is an unsteadyflow of a decaying vortex, which has an exact closed form solutionof the incompressible Navier–Stokes equations in Cartesian coordinates. The flow starts from an initialcondition given asu=Acosaxsinbysincz ,(5)v=Bsinaxcosbysincz ,(6)w=Csinaxsinbycoscz ,(7)p=p0.(8)over a domain 0≤x, y, z≤2πperiodic in allx,y, andzdirections, andp0is initial pressure constantover the whole domain.A simplified version is defined on a 2D domain, i.e. assumeC= 0 andcz=π/2. Since the vortexdecays, we can assume that velocity components decay exponentially by time, i.e. we could assume:u(x, y, t)=˜u(x, y)e-ct,(9)v(x, y, t)=˜v(x, y)e-ct,(10)where the coefficientcis yet to be defined. Solve incompressible Navier-Stokes equations and findu,v,andpas a function of time and space. To do so, you need to find or guess what ˜uand ˜vcan be.Hint: Seek inspiration from initial condition.References D Quemada. A rheological model for studying the hematocrit dependence of red cell-red cell and red