CFD

What are the advantages/disadvantages of a tetrehdral mesh?

*Advantages* - Arbitrary volume can always be filled with a tetrahedra - Can be generated quickly, automatically and for complicated geometry - Can be easily combined with curvature and proximity size functions to automatically refine the mesh in critical regions - Can be combined with inflation to refine the mesh near solid walls (boundary layer resolution) *Disadvantages* - Element and node counts are higher for a hex mesh with a similar density - Generally not possible to alight the cells with a flow direction - Not well suited for thin solids or annul due to non-isotropy of geometry and nature of element

Describe Direct Numerical Simulation (DNS)

*Direct Numerical Simulation (DNS)* - 3D transient solution : Mesh everywhere fine enough to resolve the smallest turbulent eddy :Time step small enough to resolve the fastest turbulent fluctuation frequency - No modification of the equations to be solved :High-order spatial and temporal discretisation to ensure numerical diffusion is much lower than turbulent eddy dissipation - Very highly detailed results, allowing fundamental investigation of the structure and effect of turbulent flow - Extreme computational cost, meaning DNS so far only suitable to low Re flows

Give examples for elliptic, parabolic and hyperbolic equations

*Elliptic* - Equilibrium problem - Heat equation (d^2F/x^2 + d^2F/dy^2 =0) *Parabolic* - Marching problem - Transient (unsteady) problems, all wave problems (including shocks) - 1D diffusion equation (dF/dt = v*d^2F/dx^2) * Hyperbolic* - Marching problem - Transient problems with little or no diffusion - 1D wave equation (d^2F/dt^2 = c^2*d^2F/dx^2)

Discuss the domains of dependence for elliptic, parabolic and hyperbolic equations

*Elliptic* A disturbance in the centre of the domain affects the solution everywhere (signals travel in all directions) *Parabolic* A disturbance in the domain can only affect the solution at later times (hence the marching behaviour in time), but spreads out in all directions in space *Hyperbolic* - A disturbance in the domain can only affect a limited region of the domain - Disturbance information travels through the domain at a finite speed (wave speed)

What are some other methods of flux vector splitting?

*Flux Difference Splitting (FDS)* • FDS still based on wave speeds • Performance is comparable, difference between FDS and FVS not relevant to this course *Advection Upstream Splitting Method (AUSM )* - Also called Liou-Steffen flux vector splitting - Much simpler splitting, not based on wave speeds but instead on Mach number alone • Does not require computation on the flux Jacobian, so can be faster per time step - Also separates out the pressure from the rest of the flux vector

What are the guidelines for mesh quality requirements for fluent?

*Skewness* - Hex/Tri/Quad = <0.8 - Tetra: <0.9 *Aspect Ratio* - <40 but depends on flow characteristics - More than 50 may be tolerated in the inflation layers *Cell Size Change* - between 1 and 2

What is the difference and limitations on the Peclet (Pe) Courant number (Co) and diffusion number (D)

*Steady Flows* Peclet number (Pe) = F/D=rho*u/(gamma/deltax) *Unsteady Flows* Courant (Co) = u*dt/dx < 1 Diffusion Number (D) = gamma*dt/rho*dx^2 <0.5 Note: no numerical stability requirement for implicit scheme

Compare/contrast the k-ε and k-ω turbulance models

*k-ε (k-epsilon)* - Most common turbulance models - Shown to be useful for free-shear layer flows with relatively small pressure gradients.' - for wall-bounded and internal flows, the model gives good results only in cases where mean pressure gradients are small - Model would be an inappropriate choice for problems such as inlets and compressors *k-ω (k-omega)* - shear stress transport (SST) formulation combines the best of two worlds - use of a k-ω formulation in the inner parts of the boundary layer makes the model directly usable all the way down to the wall through the viscous sub-layer, hence the SST k-ω model can be used as a Low-Re turbulence model without any extra damping functions. - Switches to a k-ε behaviour in the free-stream and thereby avoids the common k-ω problem that the model is too sensitive to the inlet free-stream turbulence properties - merit it for its good behaviour in adverse pressure gradients and separating flow - model does produce a bit too large turbulence levels in regions with large normal strain, like stagnation regions and regions with strong acceleration

What are the two equation turbulance models

*k-ε (k-epsilon)* • Transport equations (PDEs) for turbulent kinetic energy k and dissipation ε • Standard, Renormalization Group (RNG), Realizable versions *k-ω (k-omega)* • Transport equations for turbulent kinetic energy k and turbulence frequency ω = ε/k • Original Wilcox, Menter Shear Stress Transport (SST) versions

Discuss Iterative convergence error

- Caused by not driving the residuals to low enough values - Fluent uses residuals normalised against the mass residual at the start of the simulation - For steady flows, default is to reduce all residuals by 3 orders of magnitude • This may not be enough for some problems • Alternatively, this may not be achievable if the starting guess was a very good one • Should always monitor some additional measure to decide when to stop iterating, e.g. mass flux imbalance between inlets and outlets - For unsteady flows, a good rule of thumb is to reduce residuals by 2 orders of magnitude at every time step

Discuss Roundoff Error

- Computer represents numbers with finite number of significant figures - Subtraction of two similarly-sized numbers can lead to loss of significant figures - In many low speed flows, pressure variation is small • Calculation of pressure gradients requires subtraction • Use of an "operating pressure" to limit the effect of roundoff errors (calculations performed with gauge pressure instead of absolute pressure) - Can be reduced by use of double precision floating-point arithmetic instead of single precision in the solver • Requires more memory

What are the advantages and disadvantages of Hex Mesh

- For same resolution of flow physics, has less than half the amount of nodes as the tet-mesh - Automatic generation for complex geometries - Can result in more cells being required than for a stuctured hexahedral mesh

Describe the Artificial Compressibility" method as a density based solution method

- In Navier-Stokes equation, density appears as a constant - The artificial compressibility method introduces additional pseudo-time-derivatives, and then marches in pseudo-time until steady state is achieved - At steady state (in pseudo-time τ), the pseudotime derivatives are zero, so the original equations are recovered - β is the artificial compressibility factor, can be tuned for fast convergence (like pre-conditioning) - These equations treated like the compressible flow equations (e.g. flux vector splitting)

How do the Euler equations change with incompressible flows?

- Information travels instantaneously both upstream and downstream - The equation has become elliptic in space - Central differencing is now the appropriate numerical method for spatial derivatives

Discuss Discretisation error

- Lack of sufficient terms in the Taylor series expansions - Due to discretisation of both space flux and derivative terms, and time derivative terms - In principle, can be reduced to arbitrarily small values by refining the mesh (reducing Δx) and using small time steps (reducing Δt) • In 3D, halving Δx results in 8 times as many cells, so can quickly run out of memory and computational power

Describe Reynolds-Averaged Navier Stokes (RANS)

- Method of choice for most applications - Does not simulate turbulence directly, rather its effect on the mean flow is modelled - Can be used in 2D or 3D - All flow quantities considered as a "mean" (dashed) and a "fluctuating" component around that mean representing the turbulence :"mean" flow not necessarily constant, so perhaps better to think of it as a "slow" variation, with a "fast" fluctuation superimposed - Extra terms known as "Reynolds Stresses". All RANS turbulence modelling is about predicting these stresses

What are the advantages and disadvantages of implicit schemes?

- Much larger, or no, stability limits on time step - Much larger time steps allowed, can dramatically shorten overall solution times - More complex to program - Many more calculations per time step since a matrix or sub-iterative solution required at every time step - Typically more numerical diffusion than explicit methods for the same time step size - If larger time steps are used, this means larger truncation error • Not a problem if only the steady solution is important - Use for steady problems, or for unsteady problems where the time step limit of explicit methods is too restrictive

Discuss Scheme error

- Non-physical results due to the scheme or algorithm • Wiggles in flow variables in regions of large gradients • Results that violate the 2nd law of thermodynamics - "Expansion shocks" - "Carbuncle phenomenon"

Describe the PSIO scheme as a Pressure-based solution method

- Pressure Implicit with Splitting of Operators (PISO) - Extension of SIMPLE with an additional corrector step - For steady flows, run as an iterative method - For unsteady flows, the second correction is usually considered to give sufficient accuracy and the time step is advanced without further iteration - PISO is recommended over SIMPLE for unsteady flows in Fluent User Guide

What are the elements in validation (2)

- Quantification of input uncertainty • Sensitivity analysis of input uncertainty (e.g. distance from the model to the boundaries, turbulence levels in the inflow boundary conditions) - Quantification of physical model uncertainty • Comparison of results with high quality experimental results (where the experimental setup can be precisely matched in the CFD)

What are the elements of verification? (3)

- Quantification of the effect of roundoff error • Compare single and double precision results - Quantification of the effect of iterative convergence error • Compare results with different levels of residual convergence criteria - Quantification of the effect of discretisation error • Demonstrate reduction of discretisation error by two or three levels of grid and/or time step refinement • Performance of a Richardson Extrapolation to predict the "discretisation-error-free" solution

Describe Large Eddy Simulation (LES)

- RANS does not directly simulate any turbulent eddy scales :Cannot cope with situations outside the narrow range of conditions for which each model was created - DNS simulates all turbulent scales :Impractical for most flows of interest due to the computational cost - LES conceptually halfway between DNS and RANS : Directly simulate the largest turbulent scales that carry the most energy : Model the smaller scales at which this energy is dissipated : 3D simulations Approach is reminiscent of RANS - Turbulent scale cut-off achieved with a spatial filter rather than a temporal average - Unresolved eddies known as "sub-grid scales" (SGS) - Similar effect on the Navier Stokes equations • Filtering process results in three additional sets of SGS stresses which must be modelled (Leonard, cross and LES Reynolds stresses) • As turbulent eddies are directly simulated, approach must by definition be 3D and time- dependent • Nowhere near the computational cost of DNS, but still much higher than RANS (although perhaps only several times the cost of RSM) • Still not established as a commonplace engineering tool • Other "scale-resolving simulation" (SRS) models - e.g. Detached Eddy Simulation (DES), hybrid application of LES and RANS

Describe the SIMPLE scheme as a Pressure-based solution method

- Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) - Iterative approach at each time step • Guess a pressure field p* (e.g. at the start of the iteration, use the pressure from last time step) • Solve the momentum equations using the guessed pressure field to give estimates for the velocities u* and v* (here again thinking in 2D) • Solve for a pressure correction p' (so that p = p* + p') from the momentum equations using u* and v* and using the continuity equation as a constraint - Knowing the pressure correction also allows velocity corrections u' and v' to be calculated • Update the pressure and velocities, typically with under- relaxation factors ωp, ωu, ωv to ensure stability - pnew = p* + ωpp', unew = u* + ωuu', vnew = v* + ωvv' • Check for convergence of the iteration (corrections → 0) - If insufficient, set p* = pnew and continue iterating - If sufficient, move to the next time step

What are the advantages and disadvantages of explicit schemes?

- Simple to program - Each time step is fast to compute - Stability limits in the time step, can be very restrictive - Thus many time steps required, so typically long solution times - Use for truly unsteady problems where the physical time scales are small so that a small time step is required anyway

What are the most important mesh matrics for Fluent?

- Skewness - Aspect Ratio - Cell size change

What is preconditioning in relation to low mach number flows?

- Solve a modified set of equations in which the "flow speed" and the "sound speed" are much closer together • Inverted commas used here because the variables being solved for are no longer physical quantities - Convert back from these modified equations to the physical flow variables

What are some of the reasons to use CFD techniques?

- Some environments are very hard to test in, or very expensive to get to - Some environments are delicate, hard to measure in (or potentially fatal) unless non-invasive techniques can be developed - Experimental measurement can disturb the flow and alter the parameter you are trying to measure - Experiment may not model the real world either, or require compromises and interpretation of the results - CFD techniques may allow us to do things we can't do in the real world - CFD techniques may allow many potential designs to be considered quickly and cheaply

Describe the SOR scheme as a Pressure-based solution method

- Successive Over Relaxation (ω > 1) - choice of optimum relaxation factors can greatly increase convergence rate, but this is dependent on both the flow and the grid - If convergence is poor (very slow, oscillatory or even divergent), experiment with changing the relaxation factors - Lowering these factors will typically improve stability at the expense of convergence rate

List the applications for CFD (8)

- Vehicle Aerodynamics & Hydrodynamics - Building and other structures - Internal combustion and gas turbine engine combustion - Power generation, pumps, turbomachinery - Heating and cooling - Industrial process engineering - Weather prediction and climate modelling - Biological and biomedical engineering

What are the sources of error in CFD?

-Numerical error: roundoff error, iterative convergence error, discretisation error, scheme error - Coding Error: mistakes or bugs - User error: incorrect usage of software

What are the 10 steps in the CFD modelling process?

0. Preliminary analysis of the problem **Problem Identification** 1. Define your modelling goals 2. Identify the domain you will model **Pre Processing and Solver Execution** 3. Create a solid model to represent the domain 4. Design and create the mesh (grid) 5. Set up the physics 6. Define solver settings 7. Compute and monitor the situation **Post-Processing** 8. Examine the results 9. Consider revisions to the model 10. Application of results in explaining natural observations and improving engineering designs

CFD is an interdisciplinary area that involves.... (4)

1) Mathematic description of problems, 2) Numerical methods for equations 3) Applications of the numerical methods to solve problems, and 4) applications of the new findings to improve our life environment

What is the effect of low mach number flows on Courant number?

Courant number must be chosen based on the largest wave speed - Results in a very small time step - Since the flow moves with velocity u, it thus takes a very long time for the flow to "go anywhere" - This requires very long simulation times to see any effective change in the flow

What is the difference between explicit, implicit and semi implicit equations?

Explicit: dependent variable ^n Implicit: dependent variable ^n+1 Semi implicit: dependent variable (1-a)X^n + a*X^n+1

What kind of problems can CFD solve outside aerodynamics?

Fluid flows Heat Transfer Structural Dynamics Chemical Reactions

What is the difference between fluid viscosity and turbulent viscosity

Fluid viscosity - property of the fluid Turbulent viscosity - momentum exchange due to turbulant motion

What are two options for meshing close to the wall?

Shear stress governed by velocity profile at the wall (could be object or domain wall) - Create an "inflation layer" (Ansys terminology) that resolves the flow all the way down to the wall, and gives y+ for the first cell height of 1 or 2 • Use a coarser mesh near the wall, that results in y+ for the first cell height of 30 - 200, and allow "wall functions" to predict the shear stress at the wall (based on slope of the log layer profile, fairly similar to the slope of the viscous sublayer at the wall)

Why is geometrical mesh quality important?

Sources of discertisation error - non-orthogonalty intorudces errors in flux approximations - large mesh expansion introduced errors in storage and source approximations Amplification of discretisation error - corrections to reduce errors cause by non-orthogonality can create unphysical influences Difficulties solving lineraised equations - large aspect ratios require use of more significant digits (i.e use of double precision solver)

What is the difference between validation and verification?

Validation - "Solving the right equations" Verification - "Solving the equations right"

What happens when viscosity is added to the Euler equations?

they become the Navier-Stokes equations • When the flow is viscous, boundary layers will form on solid bodies - There thus must be subsonic regions, even in a supersonic flow - The method must be robust enough to cope with both subsonic and supersonic regions (and the attending directions of information travel) • No real change to the solution method for the Euler equations - Convective fluxes E are typically upwinded as before - Diffusive fluxes L are discretised using central differences

Describe the SIMPLEC scheme as a Pressure-based solution method

varies very slightly in the calculation of the velocity correction terms u' and v' from the pressure correction p' - SIMPLE is the default for steady problems in Fluent - Fluent User Guide notes that SIMPLEC can accelerate convergence in particular problems where the pressure-velocity coupling is the primary deterrent to obtaining a solution - in practice, give it a go and see if it improves convergence

What is flux vector splitting?

• Flux Vector Splitting (FVS) is one way to correctly apply the numerical upwinding process - Use upstream differences for parts of the flux vector that contain downstream moving waves - Use downstream differences for parts of the flux vector that contain upstream moving waves • Split the flux vector into - E+ contains the positive wave speeds (moving downstream) - E- contains the negative wave speeds (moving upstream) - There are infinitely many ways to do this splittng NOTE: a very handy feature of the flux Jacobian matrix - Multiply the Jacobian by the flow variable vector, and you get the flux vector (E = JQ)

Compare and contrast pressure based and density based solution method

• Pressure-based and density-based methods are typically equally effective for incompressible flows • Density-based methods are more common for compressible flows, but pressurebased methods can also be used - SIMPLE was originally conceived for compressible flows