1、一 三维时间相关可压缩流的 Navier-Skokes 方程：(1) (向量形式)zTySxRzGyFxEtU式中，ewvuUupevE2vpewuvF2wpeuvG2xxzyxqwvuR0yyzxzyxqwvuS0zzyxzxqwvuT0其中, , zx23 xyxy, wuyvy zuwzx, yvxzz23 yvzy, ,TkqxTkqTkqz为了方程封闭，必须引入 4 个关系式(9 个自变量 ):kTpewvu,1, 3, 221wvupe 72.0Prkcp2, 4, 231CTRTThree-Dimensional Navier-Stokes equationsThree-dime
2、nsional Reynolds averaged Navier-Stokes equationsLarge viscous regionsThin shear layer approximationParabolized Navier-Stokes equationsViscous-inviscid interation modelsInviscid models Boundary layer approximationDistributed loss modelTime-dependent Euler equationsSteady-state rotational modelsPoten
3、tial flow modelsSmall-disturbance potential equation,TpkinputFluid constitutive equationTurbulence modelsNoYesNAVIER-STOKES EQUATIONS- Coupled system of five non-linear differential equations of second order, in space and time- Describe conservation of mass, momentum and energy- Describe wave propag
4、ation phenomena damped by viscosityEULER EQUATIONS- Coupled system of five non-linear differential equations of first order, in space and time- Describe conservation of mass, momentum and energy- Describe wave propagation (convective) phenomena POTENTIAL EQUATIONS- SINGLE second-order non-linear dif
5、ferential equation- Describe conservation of mass and energy- Momentum conservation not fully satisfied in presence of shocksViscous effects negligibleIsentropic, irrotational flowsClassification of various flow modelsFOR EACH ELEMENT OF FLUID：Conservation of mass Continuity EquationNewtons second l
6、aw Euler Equationof motion Navier-Stokes EquationsConservation of energy Energy EquationEquation of stateVelocity Distribution : Pressure : Density : Temperature : ,xyztpu,wxyzt,vztTDeduce flow behavior : flow separation: flow rates: heat transfer: forces on bodies( skin friction, drag, lift ): effi
7、ciencies( turbine, diffuser )Solve the equations plus boundary conditionsOverview of computational fluid dynamicsReal worldPhysicsNumerical simulationApproximation levelsDynamical approximationSpatial approximationSteadiness approximationMathematical modelDiscretization techniquesResolution of discr
8、ete system of equationsSpace discretizationMesh definitionEquationdiscretizationDefinition of numerical schemeComputational Models计算流体力学不同发展阶段所求解的四种基本方程1. 线性小扰动方程 （6070 年代） 0)1(zyxM式中 wvu , ,2. 全位势方程 （70 年代中80 年代初） 022 111 222 zxyzxyzyx aaa 3. Euler 方程 （80 年代） 0zGyFxEtU4. 平均 N-S 方程 （90 年代） zTySxRzyFxEt 年代 机型 CPU 时间 费用若在 60 年代，在 IBM704 上工作，需要 20 年，费用$1000 万在 DEC2000/500 上模拟三维机翼的绕流场（速度约为 1000万次）70 VAX750 20 小时 $1000080 IBM3033 20 分钟 $100090 Cray-2 20 秒 $10网格点数 所需 CPU 时间 所需内存小扰动方程 10 万 半小时 1MB全位势方程 10 万 3 小时 4MBEuler 方程 20 万 3 昼夜 32MB平均 N-S 方程 200 万 3 个月 256MB1M1求解二维翼型粘性绕流 1M三维机翼绕流场的数值模拟
