成果40.流體力學(xué)numerical methods for heat,年-peakchina l

1、Lecture 27: Boundary Conditions Introduction to Fluid FlowLast Time Examined physical rationale for limiter functionsApplied to multidimensional situations and unstructured meshesExamined how to implement higher-order schemesThis brings us to the end of the higher order scheme discussion This TimeLo

2、ok at boundary conditions specific to convection-diffusion problemsStart looking at the computation of fluid flow Recall: Steady 2D Convection-Diffusion Governing equation:Assume flow field knownAssume Cartesian structured meshDiscretizationAs usual, integrate over control volumeApply divergence the

3、orem, linearize source term:Discretization (Contd)Area vectors:Flux on east face:Flow rate on east face:Diffusion TermWrite as usual assuming linear profile between (E,P) etc:Face Value of For UDS, write face value as: For the higher-order schemes we studied: Discrete EquationHereterm Note extra flo

4、w rate in aPHigher-order contribution needs iterationBoundary Conditions: Inflow BCConsider a boundary where flow is entering the domainInflow BCAt the boundary, we have both convection and diffusion:Known valuesDiffusion term same as for Dirichlet BCOutflow BCFlow points out of domainTypically we w

5、ould not know on the outflow boundaryValue determined by what is happening inside domainIgnore diffusion on outflow faceInfinite face Peclet number assumption Use first-order upwind schemeOther faces of near-boundary cell treated like all other interior facesOutflow BC (Contd)Set boundary diffusion

6、flux to zero:Thus, boundary flux is:Using first-order upwind scheme:When Can Outflow Condition be Used?Outflow bc cuts off domain from downstream locationsTypically this is true only if convection dominates diffusionPef 1 Boundary should not cut through recirculation regionsGeometric BoundariesTypic

7、ally “natural” boundaries of domainWallsOn wallsNo convective fluxBoundary flux has only diffusion:Can have Dirichlet/Neumann/mixed bc just as in pure diffusion problemsIntroduction to Fluid FlowSo far, we have assumed that the fluid flow field is knownWe now turn to the computation of the fluid flo

8、w Navier-Stokes equations are special case of convection diffusion equation with = u, v or w, and = and appropriate SSo why the fuss?Steady 2D Navier-Stokes EquationsCan write Navier-Stokes equations as:For Newtonian fluid, can confirm that:Why The Fuss?NS equations are non-linearNot by itself a pro

9、blemCan do Picard iterationMore than one momentum equation (x,y,z)Not a problemCan solve each sequentiallySource terms?Stress tensor sources can be computedMain issue is that pressure is not known!Must find a way to find itWhat PDE to use?Issues with Pressure ComputationTwo different problems Where

10、to store pressure vis-vis velocityHow to find an equation for pressureIn 3D, we have 4 equations in 4 unknowns3 velocities, 1 pressure3 momentum equations and one continuity equationsIn principle, possible to solveVorticity-Stream Function FormulationsUsed during 70s-80sDerive one PDE for stream fun

11、ction and one PDE for the single component of vorticity in 2DEliminates pressure as variableOnly 2 PDEs as opposed to 3 for primitive variable formulations using (u,v,p)However, no stream function definition in 3D flowsVorticity-vector potential formulations in 3D6 components (3 for vorticity, 3 for

12、 vector potential)Fewer components (u,v,w,p) for primitive variablesDifficult to derive bcs Primitive variable formulations preferred todaySequential SolutionOur approach so far has been to solve sets of coupled PDEs in a sequential mannerWe identify:u-momentum equation for u velocity componentv-mom

13、entum equation for v velocity componentContinuity as PDE for pressurePossible solution loop:Discretize and solve u over domain, assuming v, p known from current iteratesSimilarly solve v from discrete v-mom equationsSimilarly solve p from discrete continuityContinue until convergenceEquation for Pre

14、ssureContinuity equationFor compressible flows, we could convert this into an equation for pressure. For a perfect gas, for example:What about pressible flows?No relationship between pressure and densityNo obvious way to introduce pressure into continuity equationSequential vs. Direct SchemesImporta

15、nt to recognize that the problem of pressure computation for pressible flows is an artifact of sequential solution procedureOnly for sequential procedures is it necessary to identify a PDE for each unknown variableu-mom for u velocity; v-mom for v velocity, continuity eqn for pressure; energy equati

16、on for temperature etc.For direct solution procedures, no such one-on-one identification necessaryDirect SchemesDiscretize u,v momentum equations and continuity at all cellsAssemble a big matrix:Solution vector is:No need to identify which unknown gets solved using which PDEUnfortunately too expensive for practical use as of this