Implementation of 3D Incompressible N-S Equations. Mikhail Sekachev

Transcription

1 Implementation of 3D Incompressible N-S Equations Mikhail Sekachev

2 Navier-Stokes Equations The motion of a viscous incompressible fluid is governed by the Navier-Stokes equations u + t u = ( u ) 0 Quick Facts: u = 1 2 ρ P + ν Claude-Louis Navier Sir George Stokes The advection term is NON-LINEAR The mass and momentum equations are COUPLED (via the velocity) The pressure appears only as a source term in the momentum equation(up to f(t)) There is NO evolution equation for the pressure (4 equations and 4 unknowns) The continuity equation acts as a constraint for the velocity field. Mathematicians have not yet proven that in three dimensions solutions always exist, or that if they do exist, then they do not contain any singularity. The Clay Mathematics Institute has offered a $1,000,000 prize for a solution or a counterexample. [1] u [1] Millennium Prize Problems, Clay Mathematics Institute, retrieved

3 Boundary Conditions The solid walls are always a problem! A typical boundary condition is Dirichlet: u = u x Ω b ( ) Ω,t where Ω is the boundary of the domain Ω occupied by the fluid and x Ω Ω There is no B.C. for the pressure on no-slip boundaries.

4 Initial Conditions Specification of the velocity field at the initial time: Integrating the continuity equation over Ω and using the divergence theorem yields: where n denotes the outward unit normal to the boundary Ω The initial velocity field Also u b u u0 and u 0 n u t=0 = n u b ( x) dω = 0 u 0 u0 = 0 is assumed to be solenoidal, i.e. (Solenoidal condition) satisfy the compatibility condition b = n u t= 0 0 Ω (Compatibility condition)

5 Compatibility Condition The solenoidal and compatibility conditions are necessary to prove the existence and uniqueness of Euler equations (non-steady, 2-D, incompressible, zero viscosity) Hence these conditions are independent of the viscous/nonviscous character of the fluid These conditions for the initial velocity allow for an optimal choice of the linear space for the initial data (L.Quartapelle, 1993) Therefore the compatibility condition affords solution with a minimal regularity as in most numerical schemes relying on spatial discretizations of local type

6 LBB Condition The aforementioned considerations yield LBB condition The LBB condition, also known as inf-sup or div-stability condition, states that for a given pair of approximating h h functional spaces S, if there exists a positive real 0 andv0 number γ > 0 such that inf then the discrete solutions stable. (P. T. Williams, 1993) sup h h 0 q S h h 0 ( h h, ) i q γ b v v i h i 1 0 v V h 0 h h h h ui V0 and P S0 will be Failure to satisfy LBB condition often results into oscillations in the pressure field, usually termed spurious pressure modes LBB: Ladyzhenskaya(1969), Babushka(1973), Brezzi(1974) q 0

7 Mass conservation equation The mass conservation equation acts as a differential constraint on the solution to DP+DE In the nonisothermal case there are five equations with five unknowns (namely Ux, Uy, Uz, P, T) There is NO equation for determining the pressure distribution. Major Approaches for addressing this issue are: Vector field theory (convert DM to a vector potential) Problems arise with 3-D or pressure-driven flows. Projection methods WILL handle those problems

8 The Family of Projection Methods The origin of the Family of projection methods (also called fractionalstep, or splitting methods) dates back to Originators: Alexandre Chorin and Roger Temam. The method is based on the Helmholtz decomposition (sometimes called the Helmholtz-Hodge decomposition) Vector field solenoidal+irrotational The whole spectrum of projection methods can be tentatively categorized into three major groups:** Pressure-correction methods Velocity-correction methods Consistent splitting methods **An overview of projection methods for incompressible flows. Guermond, J.L., Minev, P. and Shen, Jie. 195, s.l. : Comput. Methods Appl. Mech. Engr., 2006, pp

9 Taxonomy of INS Algorithms Based on Projection From the course materials: ES 552 Computational Fluid-Thermal Analysis. Instructor: Prof. A. J. Baker

10 CONTINUITY CONSTRAINT METHOD 1. Use-implicit Taylor series approach 2. Replace the time derivatives using momentum equations 3. Integration of these equations with a guessed pressure field results in a velocity field that does not in general satisfy the continuity constraint.

11 4. A computable strategy must rely on the accumulation of φ solutions, i.e., 5. With the Poisson equation for φ: 6. When the genuine pressure, p, is required it is assumed computable from the pressure Poisson eqn.:

12 Consistent Splitting Scheme 1. The equations are expressed in primitive variables: 2. By taking the L2-inner product of the momentum equation with q and noticing that we obtain 3. The standard consistent splitting scheme written for the forward Euler time marching scheme and first order extrapolation for the pressure is expressed as:

13 4. The accuracy of the above splitting scheme can be improved by replacing by, leading to the following algorithm: 5. To avoid computing explicitly in the second step, we take the inner product of the first equation with and we subtract the result from the second equation. 6. Owing to the identity and denoting we have the following algorithm:

14 7. Finally, replacing the advection term with its skew-symmetric counterpart the scheme corresponds to solving the following set of governing equations: All six equations in are linear, with the last one being simply an algebraic update.

15 The Weak Form The weak form of the equations above can be established using Galerkin s FE implementation, also referred to as Galerkin Weak Statement. The momentum equation in x-direction has the following ingredients:

16 The weak form for the variable φ is expressed as: The weak form for the intermediate variable Π is written as: Finally, the week form for the corrected pressure is:

17 Benchmark: Lid-driven 3-D cavity flow using consistent splitting scheme (i) The local eddies are located in the region close to the corners and near the endextremities of the cavity and are associated with fluid transport in the spanwise direction. (ii) The global eddies are regularly distributed along the cavity span in planes perpendicular to the lid and stretch along the streamlines of the primary recirculating flow. They are usually referred to as Taylor-Gortler-like (TGL) vortices.

18 Numerical results Y.-H. Kuo, K.-L. Wong, J. C.-F.Wong/ Adv. Appl. Math. Mech., 6 (2009), pp Re=3200, 1x1x1 cavity, Δt=0.005 Three pairs of symmetric TGL vortices

19 Benchmark: Thermal Cavity Ra 6 = cavity Non-uniform mesh

20 Future work/challenges Thermal Cavity and Lid-Driven cavity on So far only is working mesh Spalart-Allmaras one equation model LES Smagorinsky Subgrid-Scale (SGS) model Rational LES (RLES)