• Examining the role of the pressure, it turns out that the spherical part of the stress tensor, extracted in the pressure term directly relates to the above condition on the velocity. Viscous stress is modelled on the basis of the velocity gradient:
σ= −pI + µ£
∇u + (∇u)T¤
, (10.5)
postulating the equivalence between the mechanical and thermodynamic pressure. Therefore, the pressure term is related to the tr(∇u) = ∇•u, which appears in the continuity equation. In other words, pressure distri-bution should be such that the pressure gradient in the momentum equation enforces the divergence-free condition on the velocity field.
• If the pressure distribution is known, the problem of pressure-velocity cou-pling is resolved. However, it is clear that pressure and velocity will be closely coupled to each other.
10.2 Density-Based Block Solver
Density-Based Algorithm
• In previous lectures, we have shown a block coupled form of the density-based flow solver. Noting that all governing equations fit into the standard form and all variables are fully coupled, the compressible Navier-Stokes system can be written as:
∂U
∂t + ∇•F − ∇•V = 0 (10.6)
where the solution variable U is:
U =
• In the above, pressure appears in the convective flux F :
F =
• Standard (Roe flux) compressible Navier-Stokes solver will evaluate F for each cell face directly from the state (U) left and right from the face, using approximate Riemann solver techniques
• Looking at the second row of the flux expression we can recognise the convective contribution and the pressure driving force (note ∇•(pI) = ∇p).
In high-speed flows, the first component is considerably larger than the second
• In the low-speed limit, a pressure difference of 3−5Pa can drive considerable flow; however, in this case, the pressure gradient will dominate. As shown before, this implies a density change of approximately 5 × 10−5kg/m3 for the mean density of 1kg/m3. Equivalent calculation for a liquid (water), would produce even more extreme result (due to the higher speed of sound)
• Equation governing pressure effects in this case is the continuity, through density transport and the equation of state. Therefore, for accurate pressure data we need to capture density changes of the order of 1 × 10−5, with reference level of 1, together with the velocity changes of the order of 1 and energy level of 2 × 105 (e = ρCvT ). Note that all properties are closely coupled, which means that matrix coefficients vary to extreme levels
• The speed of sound in general is given as
c = s
∂p
∂ρ (10.9)
Infinite speed of sound (incompressible fluid) implies decoupling between density and pressure
• As a consequence of decoupling, density-based solver cannot handle the the incompressible limit. In practice, very low Ma number flow can be achieved, either through matrix preconditioning or by introducing artificial compressibility
Explicit and Implicit Compressible Flow Solver
• Relationship that prescribes F as a function of UP and UN is complex and non-linear: calculating characteristic wave speed and propagation. It is therefore natural to evaluate the flux F and advance the simulation explic-itly:
Un= Uo− ∆t(∇•F − ∇•V ) = Uo− ∆tR (10.10) Here, R is the convection-diffusion residual residual (A higher-order time-integration technique may also be used)
• This leads to a fundamentally explicit time-integration method, with the associated Courant number (Co) limit: time-step is limited by the size of the smallest cell
10.2 Density-Based Block Solver 143
• Time-step limitation is in reality so severe that it renders the code useless:
for steady-state simulations, we need to achieve acceleration of a factor of 100 − 10 000
• Solution acceleration techniques require faster information transfer in order to approach steady-state more rapidly. We will examine two:
– Implicit solver – Geometric multigrid
Solution Acceleration Techniques
• Implicit solver
– Implicit compressible solver is based on the same flux evaluation tech-nique as the explicit solver, but generalising the form of the flux ex-pression to create matrix coefficients
F = F (UP, UN) = ∂F – Here, matrix coefficient is a full 5 × 5 matrix, calculated as a Jacobian and D is the explicit correction. Linearisation may be done in several ways, with different level of approximation
A =
– With the help of flux Jacobians, we have created an implicit system of equations, which relaxes the Co number criterion, but not to the desired level. However, this is a very useful first step
• Multigrid acceleration
– Geometric multigrid is based on a curious fact: as the mesh gets coarse, the Co number limit becomes less strict, allowing the simulation to advance in larger time-steps and a steady-state solution is reached in fewer time-steps
– The problem we have solved on a coarse grid is physically identical to its fine-grid equivalent. It should therefore be possible to “solve”
the coarse-grid problem and use the solution as the initial guess for its fine-grid equivalent
– Full Approximation Storage (FAS) Multigrid performs this pro-cess on several levels simultaneously, using a hierarchy of corse grids.
This allows us to use a very large Co number (100 − 1 000 or higher) without falling foul of the Co criterion: significant part of information transfer occurs on coarse grids without violating the stability criterion – Additional complication in multigrid simulation is the requirement for a hierarchy of coarse grids for the geometry of interest. Additional problems, related to the geometric representation and specification of boundary conditions on coarse grids
– In practice, coarse grids are assembled be agglomerating fine grid cells into clusters