What are the three key advantages that make CFD a valuable tool for engineers?
Time consuming experiments can be replaced
All values can be measured in-situ
Different physical effects/aspects can be analyzed separately
What factors are essential for ensuring the applicability of a CFD simulation to real-world scenarios?
Why might this applicability not always be guaranteed?
Similarity of geometry & dimensionless numbers has to be guaranteed
Might not be possible to ensure all characteristic numbers are similar
In 1922, Lewis Fry Richardson conducted the first known flow simulation in an attempt to forecast the weather for Europe, which ultimately failed. Reflecting on this failure, what crucial aspect did Richardson overlook in his calculations? How did this oversight contribute to the failure of the simulation?
He neglected turbulence, therefore the dissipation of the system was way lower than the real dissipation. This caused the simulation to diverge
What does the acronym CFD stand for in English?
Computational fluid dynamics
Outline the three fundamental modeling steps that form the basis of every flow simulation.
Physical Modelling (e.g. Continuum assumption, Chemical reactions…)
Mathematical Modelling (e.g. Turbulence model, Boundary conditions)
Numerical Modeling (e.g. Solving algorithms, Grids / Meshes)
While CFD theoretically allows for the modeling and calculation of various phenomena, what do you perceive as the greatest challenge when simulating the launch of a rocket? Provide an illustrative example.
The launch of a rocket is a multiscale problem in space & time. Trying to calculate the combustion which happens in very short length and time scales [ms] & [mm] vs. the outer aerodynamics of the rocket & atmosphere [s] & [m], which happen on way bigger scales. Resolving this simulation accurately would require massive amounts of computational power.
The 'Periodic Hill Flow' serves as a benchmark experiment for simulation tools. Describe a) the experiment itself and b) the specific challenges associated with simulating the flow in this scenario.
The experiment consist of a hill and a valley with a specified geometry. The inflow conditions on the left are the outflow conditions on the right, therefore “periodic”. With a reynolds number high enough the flow separates after the hill. Especially the reattachment of the separation bubble is difficult to simulate. Even this simple geometry results in very different solutions of different CFD codes.
Name the three consecutive work steps for conducting a CFD simulation
Preprocessing
Create grid
Define boundary conditions
Select models
Solution
Calculate solutions
Postprocessing
Visualize results
Physical evaluation
Estimate uncertanties
How do fluids and solids differ under shear stress?
The shear stress of a newtonian fluid is proportional to the shear rate. (Theoretically unlimited)
tau ~ dalpha/dt
The shear stress of a solid is proportional to the deformation. (This is limited by the max shear stress of the material)
tau ~ alpha
The gas flow in the lung capillaries is to be calculated. By which dimensionless number can you determine whether a calculation with the finite volume method is justified?
How is it defined?
Which alternative method would you suggest if a finite volume method is not justified?
Kn = lambda/L
Kn: Knudsen number
lambda: mean free path length
L: characteristic length (in this example the diameter of the capillaries)
The knudsen number is important to evaluate whether the continuum assumptions holds true for a specific case. If Kn < 0.01 Continuum flow can be assumed. If it is greater, the fluid cannot be assumed to be continuum anymore and therefore the fluid volume method is no longer justified.
In this case one can use molecular dynamics and other particle based methods.
Which dimensionless number:
1. Describes the ratio of inertial to frictional forces in a flow? Name and formula.
2. Describes the compressibility of a flow? Name and formula.
3. Represents the ratio of inertial to surface forces? Name.
4. Plays an important role in unsteady flow separation around blunt bodies? Name.
5. Is relevant for describing a thermal boundary layer? Name.
6. Represents the ratio of inertial to gravitational forces and is therefore relevant for calculating gravity waves? Name.
Reynolds number: Re = U*L/nu
Mach number: Ma = U/a
Weber number
Strouhal number
Prandtl number
Froude number
Given the total derivative of the quantity φ(x,t), what is the difference compared to the material derivative?
Why is the material derivative used more often in fluid mechanics?
The material derivative is the total derivative with dx/dt being the flow velocity. It describes the fluid in a lagrangian way.
Conservation laws for a unit mass have a Lagrangian form -> material derivative more common
Given is the material derivative of the quantity φ(x,t). Mark the local and convective terms.
Which term becomes zero in the case of a steady flow?
First: Convective
Second: Local
For a steady flow, the second term becomes 0 (no change in time).
How many unknowns must you determine when calculating a compressible 3D flow?
Which equations are available for this purpose?
u,v,w,p,T,rho (6)
Three momentum equations
One continuum equation
One energy equation
One equation of state
What describes the mass conservation of an incompressible fluid?
Divergence of the velocity field = 0
Which term of momentum conservation approaches zero in creeping flows?
What does this mean for the equation to be solved?
The convective term approaches zero
The dissipative term becomes dominant
Given the momentum conservation in integral form:
- Mark with ① the dominant term in the case of a creeping flow and with ② the term that can most likely be neglected.
- Mark with ③ the dominant term in the case of a flow with Re → ∞. Mark with ④ the term that can be neglected.
Which term of the Navier-Stokes equations is neglected in deriving the Euler equations?
What does this mean for the flow?
Euler equations are the inviscid navier stokes equations “Re->inf“-> no dissipative term
Given the momentum conservation in differential and integral form. Name one application for each
The differential formulation is used to define the finite difference method
The integral formulation in its conservative form is used to define the finite volume method
The right-hand side of the momentum conservation corresponds to the sum of acting forces.
Provide one example each for a point force, surface force, and volume force.
Point force:
Body forces…
Surface force:
Pressure, friction, surface tension
Volume force:
Gravity, centrifugal forces…
Name three flow mechanically relevant effects that a turbulent boundary layer has compared to a laminar boundary layer
Turbulent boundary layers:
thicker
higher heat transfer
higher friction
improves mixture
Give three fundamental properties of turbulent flows
always 3-D
chaotic
viscid
instationary
coherent structures
rotational
Explain the term "turbulent energy cascade”
Provide the basic properties of:
large scales of a turbulent flow
small scales of a turbulent flow
Large scales:
Generated by external energy supply
Dependant on geometry and boundary conditions
Large and long-living
High-energy
Very difficult to model
Models are always made for special cases
Small scales:
Generated by cascade process from large eddies
Relatively universal
Often quite homogenous and isotropic
Small and short-lived
Low-energy
Easier modelling
Universal models possible
Given the kinematic viscosity ν [m²/s] and the energy dissipation rate ε [m²/s³].
Determine the Kolmogorov length and Kolmogorov time.
Gives an estimate for the smallest vortex elements within the flow:
Gives an estimate for the smallest timescales within the flow:
Which length scale describes the size of the smallest vortices in a turbulent flow?
Kolmogorov length scale
By which method can the largest structures of a turbulent flow be determined?
Two-point-correlation
What purpose does Gauss's theorem serve in the finite volume method?
The volume integral of the control volume can be translated to a surface integral. Therefore the fluxes across the boundaries of the elements determine the internal state of the finite volume elements.
In computational fluid dynamics (CFD), the flow domain is typically divided into subunits where conservation equations are solved.
Name the three most important discretization schemes.
Finite Volume (averaged values of the cells (robust))
Finite Difference (point values on cell center or edges)
Finite Elements (coefficients of basis functions)
What is the purpose of grid generation in computational fluid dynamics?
Discretize space in order to solve equations
In which regions of the flow is high resolution particularly useful? Explain.
High gradients, boundaries…
Explain the term "quadrature" in the context of the finite volume method (why and how?)
Quadrature is the numerical approximation of any integral. In the context of FVM, the surface integrals are approximated by numerical quadrature.
This is done, since the FVM method solves the N-S equations in an integral form.
In the context of grid generation, what is the difference between topology and geometry?
Topology: The relationship between neighbouring cells
Geometry: Shape and dimensions
Name three advantages each of structured and unstructured grids.
Structured:
Normally simple data structure (e.g. 2-D / 3-D matrices)
simple topology and quick implementation
efficient algorithms applicable
allows grid generation “by hand” -> optimal result control
Very precise solutions for flows with a pronounced main flow direction (e.g. near walls) possible
Reduction of numerical diffusion
Unstructured:
complex geometries can be modeled very easily
Simple grid refinement by adding points in the area of strong gradients
(Downsides:)
Mesh refinement complex
no global structure (usually more complicated data structure)
constant access to the connectivity matrix
higher computing times
How can grid quality affect the accuracy and convergence of CFD simulations?
It can have drastic effects on both accuracy and convergence. Holes, overlapping cells.. for example may cause the numerical errors or divergence of the simulation.
A structured grid that is aligned with the main direction of the flow however can decrease the numerical diffusion
Which parameter is used to characterize the grid near walls?
What must be true for this parameter to resolve turbulent boundary layer flows?
Wall distance: y+ should be ~1
(Why?:)
There are large gradients next to the wall. Most of the turbulence production happens in y+<20
Therefore this region has to be resolved fine enough. When y+ is chosen to be ~1, experience shows reliable and reproducible results.
What is a connectivity matrix in the context of grid generation?
The connectivity matrix maps the nodes to their respective elements:
Name the three basic topologies of structured grids.
(Y-Grid)
Name three requirements for good grid quality
• The generated grid must not have any overlapping grid lines, cells must have a positive volume.
• There should be the option of increasing the density of grid points and grid lines in arbitrarily selectable areas.
• The position of the points on the boundary should be definable in order to ensure an exact reproduction of the geometry.
• Grid should be smooth.
• Orthogonal grid if possible.
• Short computing times (few cells, fast convergence, etc.)
What is the concept of "advancing front" in the context of grid generation?
Method used for unstructured grids generation.
Starting from elements along edge / boundary, the interior volume is stepwise filled with additional elements
How does the advancing front technique work in unstructured grid generation?
Specifies point distribution along edge of region, creating a layer of elements along this edge, and continually redefining the edge (the "advancing front") until the entire region is filled
What is the purpose of grid refinement in CFD simulations?
Its used to increase the resolution in areas where high accuracy is needed, such as regions with high gradients or near boundaries, by adding more and smaller cells
What is the main advantage of structured grids over unstructured grids in terms of numerical diffusion?
When is this advantage particularly significant?
Structured grids can reduce numerical diffusion by aligning grid lines along the streamlines and making the flow towards the cell surfaces as orthogonal as possible.
This is particularly beneficial for flows with a pronounced main flow direction.
Describe the term "triangulation".
What rule is prescribed for Delaunay triangulation and what advantage does it provide for grid quality?
Triangulation is the process of dividing a geometric domain into a set of triangles.
No point should be inside the circumcircle of any triangle in the triangulation.
Maximum angles, full reproducibility with given point cloud, Adaptability, easy to refine
Explain why the computational and memory requirements are higher for an unstructured grid than for a structured grid.
Complex Data Structures, Connectivity Matrix, complex solving algorithm,…
What is a multi-block grid, and how does it contribute to efficient grid generation for complex geometries?
What is a disadvantage?
MB-grid divides into subregions (blocks) with individual grid structure (mainly structured). Simplifies grid generation of complex geometries into manageable parts and allows for efficient grid generation and optimization within each block.
Disadvantage:
More complex data exchange at interfaces – often with higher numerical diffusion.
Given the following pipe cross-sections, sketch the blocking for a structured multi-block grid with
as uniform quadrilaterals as possible.
Turbulence modeling aims to massively reduce the computational cost of Direct Numerical Simulation (DNS) while accurately capturing the effects of turbulence.
a) Which scales are particularly suitable for modelling and why?
b) On what mathematical approach is the RANS modelling based?
c) On what mathematical approach is the LES modelling based?
a) Small scales, since they are often homogeneous and isotropic, relatively universal and therefore easy to model.
b) Time-averaging the flow leads to the reynolds averaged Navier stokes equations. -> It models the whole turbulence spectrum
c) Resolve the large scales and filter out the smaller ones. -> It models the small scaled turbulence.
Draw a typical spectrum of the turbulent kinetic energy of a turbulent flow. Make sure to label the axes correctly.
1. Optional: Qualitatively mark the regions resolved in DNS and LES and explain the role of the sub-grid scale model.
2. Optional: How does this spectrum change with increasing Reynolds number?
Goal of the sub-grid model: “Modeling of the impact of subgrid-scales on resolved scales as a function of the resolved scales”
When the reynolds number increases the structures within the flow generally get smaller, therefore it gets stretched to the right with LES needing to resolve lower wavenumbers
What is the prerequisite for a direct numerical simulation (DNS)?
How is turbulence modelled in this case?
Massive computational power and a relatively low Re number
Turbulence isn’t modelled, it is fully resolved. This is the “correct solution” to the flow problem, needs massive compute however.
With which dimensionless number and which power does the computational cost of a DNS scale?
Re^3
(Kolmogorov-length: eta_k)
What do the abbreviations RANS, DNS, and LES stand for, and what is the difference in their modelling approaches?
RANS: Reynolds Averaged Navier Stokes equations (Models the whole turbulence spectrum)
LES: Large Eddy Simulation (Models small scales in the turbulence spectrum and resolves the large ones)
DNS: Direct Numerical Simulation (Doesn’t model anything)
Which term in the momentum conservation equation leads to the "closure problem" in RANS modelling? How many unknowns does the closure problem involve?
Reynolds stress tensor. Initially 6 unkowns
Which entries of the Reynolds stress tensor (RST) describe the turbulent kinetic energy (TKE)
The trace i.e. (𝑖 = 𝑗)
Mark the correct statements with "T" (true) and the incorrect statements with "F" (false):
How do the values on the diagonal and off-diagonal of the Reynolds stress tensor behave in the time average in the case of isotropic turbulence?
Into which two categories can the RANS turbulence models covered in the lecture be divided? What is the difference in their modelling approach?
Name one advantage and one example for each category.
Reynolds Stress Models (RSM): A fundamentally different approach of Reynolds-stress-models is, that transport equations are formed directly for the unknown Reynold-stress-tensor.
Eddy Viscosity: Areas of turbulence, exchange momentum and therefore cause momentum diffusion comparable to molecular viscosity. Hence, they introduce a numerical viscosity - the so called eddy viscosity.
Advantages:
RSM: Physical representation of turbulence
Eddy Viscosity: Low computational effort, since there are only one/two additional equations to be solved
Imagine you are to calculate the mean flow in a swirl combustion chamber. The adjacent image shows the geometry and mean streamlines of such a system. Choose an appropriate RANS turbulence model and justify your choice.
Swirl -> Reynolds-stress models, since they produce better results for flows with streamline curvature
What is the difference between eddy viscosity and kinematic viscosity?
Eddy viscosity is numerically introduced viscosity that doesnt exist. It tries to model the effects of turbulence, since this acts like viscosity.
The kinematic viscosity is a material property.
Where do the names zero-equation, one-equation, and two-equation model come from? Provide an example for each.
The names come from the number of additional equations used to model the turbulence
Examples:
Zero-equation: Prandtl Mixing Distance Model
One-equation: Prandtl One-Equation Model / Spalart-Allmaras 1994
Two-equation: k-ε Modell (Jones and Launder, 1972) / k-w Modell (Wilcox, 1988)
Name one eddy viscosity model, one eddy viscosity transport model, and one Reynolds stress model.
Eddy viscosity model: k-ε Modell (Jones and Launder,1972) / k-w Modell (Wilcox, 1988) / Prandtl Mixing Distance Model
Eddy viscosity transport model: Spalart & Allmaras (1994)
Reynolds stress model: BSL EARSM (Wallin and Johansson 2000)
The most well-known two-equation models are the k-ω, k-ε, and SST models. Name an application area where each of these models yields good results.
k-ω: Internal flows with boundary layers
k-ε: External flows and 2-dimensional thin shear layers
SST models: Boundary layer flows with streamline curvature
Name a significant disadvantage of the k-ε model and how it can be overcome.
The k-ε model is restricted to flows without high pressure gradients and without flow separation. Adjustment of the transport equations to the k-omega model fixes some of theses short comings.
What is the main problem with the use of wall models in CFD simulations?
Turbulence production happens largely close to the wall, therefore it has to be resolved adequately. However most of the wall models fail with curvature, pressure gradients and non-equilibrium.
Kolmogorov and Rotta simplify the dissipation term by assuming isotropy.
Why is this assumption justified?
The dissipation happens at very small scales (see turbulence spectrum), this small scale turbulence can usually be assumed to be isotropic.
The pressure-strain term is usually divided into a slow term and a rapid term. Describe how both terms work.
The slow term drives the anisotropic turbulence to be isotropic (assumption of rotta).
The rapid-term changes the Reynolds-stress-tensor due to the external forces that are caused by the mean gradients.
What phenomenon is depicted here?
Which turbulence model is suitable for simulating this phenomenon?
Corner vortex
-> RSM models can capture these phenomena!
In the diagrams below, the dimensionless velocity 𝑢+ is to be plotted against the dimensionless wall distance 𝑦+.
▪ Plot the experimentally determined profile of a turbulent boundary layer in the first diagram.
▪ Plot the profile obtained using an ϵ-based RANS model in the second diagram.
▪ Plot the profile obtained using an ω-based RANS model in the third diagram.
Which mathematical tool can be used to derive the order of a method?
Taylor series - leads to an approximation of the function, the lowest non zero term is the order of the method.
What does the order of a method indicate? Be precise!
The order determines the rate of grid convergence. Higher order methods dont necessarily have a lower discretization error however!
(E.g. doubling the grid points with a second order method leads to a decrease in the discretization error to one-quarter)
What does the midpoint rule state?
What does the trapezoidal rule state?
UDS: Provide Ψ𝑒 for
o 𝑈>0
o 𝑈<0
CDS: Provide Ψ𝑒 for an equidistant grid for
Explain the term numerical viscosity. In methods of which order does it occur?
The truncation error generally is generated by the numerical discretization scheme. This has the same character as a diffusion term and is therefore also called "numerical diffusion".
It occurs in lower order methods
Given the numerical viscosity for a UDS method, when does this method become unstable? Derive a stability criterion (CFL number)
Unstable for vN < 0:
Rearrange to CFL expression:
-> For instability: CFL > 1!
How does numerical viscosity affect the Reynolds number of the simulation?
-> The Reynolds number decreases
The sketch shows the exact solution 𝑢(𝑥) for a shock wave. Draw a numerical solution with dispersive and dissipative error behavior. Name a method that exhibits each type of error characteristic.
Dispersive: CDS
Dissipative/Diffusive: UDS
Given the truncation error E for the friction term in the momentum equation, explain why a uniform grid is better than a non-uniform grid.
For uniform grids XE-Xe = 0.5 and Xe-Xp = 0.5
-> The first term cancels out and the order of the error increases by one to a second order method
Briefly explain what the "high-resolution" scheme in ANSYS CFX does.
Name one advantage and one disadvantage.
It blends between CDS and UDS with a blend factor beta which CFX automatically adjusts locally.
Advantage:
Lower numerical diffusion
Higher computational effort (?)
For which CFL number does the numerical diffusion coefficient of the UDS method approach zero? Show this by calculation.
vN = 0
Rearrange to CFL-Expression
-> vN is zero for CFL = 1
What is meant by the term backscatter? Describe with one sentence.
It describes the effect of small scale turbulence on larger scale turbulence.
In LES, a distinction is made between large-scale (coarse) and small-scale (fine) structures.
Briefly describe how they interact with the grid size.
Coarse structures can be resolved on the given grid scales, whereas the fine structures cant. The grid already applies a certain filter, since the smaller turbulence cant be resolved and has to be modelled.
Explain the expression “well resolved LES”.
According to Pope: 80% of TKE has to be resolved
Plot on the diagram below where RANS, DNS, and LES are located for a flow with a high Reynolds number and a low Reynolds number.
For a given flow u and a given filter kernel G, how do you obtain the large-scale and small-scale structures of the velocity?
Which of the following statements is correct?
Given the filtered momentum equation. How can the subgrid-scale stress tensor be derived from it?
By comparing the filtered momentum equation and the incompressible Navier-Stokes-Equation:
Given the subgrid-scale stress tensor. Derive the Leonard, cross, and Reynolds stress tensors from it.
Given the Leonard, cross, and Reynolds stress tensors. What physical effects do these tensors describe?
Draw a diagram of the turbulent kinetic energy (TKE) – mark, which part is modelled, and which part is simulated with LES / DNS / RANS.
What is the task of a subgrid-scale model?
It tries to model the impact of the subgrid-scales on resolved scales as a function of the resolved scales
Describe the similarity between Smagorinskys approach and a class of basic RANS models?
This basic assumption of RANS k-epsilon models, is that the turbulence acts as a viscosity the so-called “eddy viscosity“.
In the smagorinsky model, the subgrid-scales are also modelled with the assumption of eddy viscosity.
Smagorinsky introduces a constant to calibrate 𝑣𝑆𝐺𝑆. What is the major drawback of this?
This “constant“ is constant within the flowfield, varies strongly among different flows and has to be adjusted to a given flow problem however.
Is backscatter possible with the Smagorinsky model? Justify your answer.
Backscatter is not possible, as (𝐶𝑆∆)^2*|𝑆𝑖𝑗| ≥ 0 (Subgrid-viscosity 𝜈𝑆𝐺𝑆 is never negative)
What procedure do Germano et al. (1990) propose with their dynamic Smagorinsky model to improve the original Smagorinsky model (1963)? What is the major advantage of this?
The Smagorinsky-constant CS=CS(x,t) is determined dynamically through a test-filter.
No additional empirical assumptions required
CD can assume positive and negative values and is therefore able to model “backscatter”
What is the main assumption of the dynamic Smagorinsky model regarding the interaction of scales?
Both scales are almost of the same size and resemble each other.
Idea: Calculation of 𝜏𝑖𝑗 from the smallest scales of the known resolved scales.
What are the advantages of the dynamic Smagorinsky model over the conventional Smagorinsky model?
Significant improvement over the Smagorinsky model
Improved rendering of near-wall turbulence
What is the biggest disadvantage of the dynamic Smagorinsky model compared to the conventional one?
In practice, this often leads to instabilities so that negative values of 𝜈𝑆𝐺𝑆 are often deleted.
The small scales arent assumed to act as a eddy viscosity. The resolved velocity field is inserted in the subgrid-scale-tensor
as an approximation of the unfiltered field
Deconvolution of the filtered velocity to a geometric series (Neumann-series)
The vSGS isnt used, since they arent based on eddy viscosity
Explain the difference between an implicit and an explicit time integration method using the example of the equation 𝑑𝜑(𝑡)/𝑑𝑡 = 𝑓(𝑡, 𝜑(𝑡))
The explicit method only uses phi at point n. Implicit methods estimate phi n+1 with the values phi n+1 itself.
Explain the midpoint and trapezoidal rules. State their respective orders
Name three advantages of explicit time integration and one disadvantage.
Efficient, only one iteration needed
Easy to implement
Low memory requirement
Unstable for long time steps
Why are implicit methods often unsuitable for LES simulations?
Implicit methods can lead to higher numerical diffusion which is not suitable for LES
Name two methods to achieve a higher-order time-stepping scheme (>2nd order) and describe the approach.
Additional nodes (‘time steps’) are required to achieve a higher order of accuracy
These can either be:
points at which the solution was determined at earlier times (tn-1, tn-2, tn-3 ,...) -> Multi-step methods
additional points between tn and tn+1, that are only used for numerical purposes -> Runge-Kutta-methods
Explain multi-step methods and name one advantage and one disadvantage.
If you want to achieve a higher order of accuracy, additional nodes need to be taken into the estimation.
For Multi-step methods these are points “in the past” where the solution is known (tn-1, tn-2, tn-3 ,...)
e.g. Adams-Bashforth method/Adams-Moulton method
Higher order methods
Other methods are necessary in order to start the simulation
Explain the Runge-Kutta method.
For the Runge-Kutta method these are points are predicted points between tn and t^n+1.
Predictor step: Predict the solution on the additional timestep(s) via an euler explicit method
Corrector step: Calculate phi^n+1 based on the predicted solution of the additional timestep(s) via a mid point rule
Explain the predictor-corrector method. What is the order of this method?
Combination of explicit and implicit methods
The solution at the new time is predicted with an explicit Euler- forward time step (predictor)
The solution is corrected by applying the predictor in the implicit trapeze rule (corrector)
-> Second order
How does the CFL number differ between compressible and incompressible flow?
What is meant by local time-stepping? Name one advantage and one disadvantage.
Local time step: CFL number is kept steady
-> Time is different in each cell (at each time step)
Convergence may be achieved way faster
Not fully converged solutions are unphysical
What is meant by physical time-stepping? Name one advantage and one disadvantage.
Physical time step: The physical time step dt is kept steady
-> To ensure stability, all cells have to be aligned with the "smallest cell"
Transient is itself a physical solution
Convergence takes long or may never be reached
Explain the terms "local time-stepping" and "global (or physical) time-stepping" and give one advantage for each.
Local time-stepping:
CFL number is kept constant, the time is different in each cell and timestep
Advantage: Convergence is way faster
Physical time-stepping:
Timestep dt is kept constant, the time is the same in each cell and timestep
Advantage: Transients are physical solutions
Why do local timestepping and physical timestepping methods yield the same converged result in steady-state calculations?
For physical timestepping this is reached at the same time in the whole flowfield
For local timestepping this is reached at different times in the flowfield
-> If the steady-state solution is reached d/dt = 0
Given the following linearization scheme.
Calculate the error and determine the order of the method.
For efficient methods, a system of equations is often created that looks as shown below. What computational step is necessary to bring the momentum conservation into this form? Why is this step necessary?
𝐴 ∙ 𝜑 = 𝑏
Linearizing the non-linear Navier Stokes equations
Necessary for implicit methods
Why is linearization frequently used?
For the application of implicit (efficient) time integration schemes
Name four boundary conditions with different physical significance.
Dirichlet boundary condition:
Dependent variable (Ψ) on boundary is given e.g. wall temperature
Neumann boundary condition:
Gradient (𝑑Ψ/𝑑𝑛) of the dependent variable is given e.g. heat flux density
Robbin boundary condition (mixed BC):
Heat flux density via heat transfer coefficient 𝑎 and ambient temperature 𝑇 is given
Periodic boundary condition:
Boundaries on opposing sides are interpreted as neighboring cells (e.g.: turbine grids, periodic hills)
How many quantities need to be specified for a supersonic inlet boundary condition?
For the Navier-Stokes equations, exactly 5 independent quantities must be specified as a Dirichlet boundary condition, e.g.:
Density, 3 x impulse and energy, or
Total pressure, total temperature and velocity vector (stable)
What needs to be specified for a supersonic outlet boundary condition and why?
It is not allowed to specify Dirichlet boundary conditions.
-> Neumann boundary conditions for all quantities.
-> At a supersonic outflow, the information about the conditions at the outflow can’t be transported into the flow. Setting a dirichlet BC here can’t adjust the flow that this BC would be satisfied
Describe the term "periodic boundary condition." Name a practical application.
„Boundaries on opposing sides are interpreted as neighboring cells in order to project spatial-periodically occurring areas on one area“ (e.g.: turbine grids, periodic hills).
What conditions apply to the flow velocity on a 2D plane no-slip wall and what can be derived from these conditions?
No-slip condition -> Velocity value must be equal to velocity of wall
No flow across wall -> Vertical velocity must be zero
Given the velocities u and v for a 2D problem in the wall-adjacent cells u_0 = 1 m/s & u_1 = 4 m/s. What values are written in the ghost cells
a) In the case of a symmetry boundary condition?
b) In the case of a no-slip wall?
Symmetry & No-slip wall
What is the purpose of the so-called "Rhie and Chow interpolation"?
How is this achieved?
The purpose is to handle unphysical velocity/pressure oscillations in the grid.
Modified CE (Rhie & Chow Interpolation - 1983):
Special interpolation rule for discretizing the continuity equation.
Integration of the pressure gradient in the calculation of the velocities on the finite-volume surfaces (ue,uw,vn,vs) for the evaluation of the source term of the pressure-Poisson equation.
What is meant by checkerboard oscillations of the pressure field? When can these occur?
These can arise, when the Pressure Poisson Equation is solved with a central differencing scheme. The momentum equations nor the conservation of mass can couple u and p at the interpolated point P. -> Unphysical oscillations can arise / remain
Given the system of equations 𝐴 ∙ 𝜑 = 𝑏 where 𝜑 is the solution vector.
The formal solution is 𝜑 = 𝐴^-1 ∙ 𝑏.
1. What properties do A and A^-1 typically have?
2. What is the memory requirement for N grid elements?
A is a sparse matrix, only having entries in its diagonal and off diagonals A^-1 is a dense matrix
For N grid points, A and A^-1 have O(N^2) elements -> N^2 memory requirement
Name two direct methods for calculating A^-1
Which order of magnitude of computational steps is required for a grid with N cells?
Gauss elimination & LU
-> O=N^3
Name two iterative methods for calculating 𝜑
Jacobi
Incomplete LU
Gauss-Seidel
What is the difference between direct and iterative methods?
Direct methods:
Exact solution with maximum „computer accuracy“
Very high effort:
High storage requirements, e.g. O(N^2)
Many computational operations, e.g. O(N^3)
No need to solve the system of equations exactly, because discretization errors are much greater than the "computer accuracy“.
Iterative methods:
Calculation of an approximate solution by iteration 𝜑𝑛+1 = 𝑓(𝜑𝑛)
Why are iterative methods preferred over direct methods? What is the rationale behind this?
They are faster. No need for computational accuracy due to other modelling mistakes with higher impact
Given the system of equations 𝐴 ∙ 𝜑 = 𝑏 . For iterative methods, the matrix A is typically split into an easily invertible part N and a hard-to-invert part P. Derive a simple iterative scheme from this.
Describe the terms "error" and "residual" of an iterative solver.
Explanation and formula
Error: Deviation from the exact solution (generally unknown)
Residuum: Deviation of the solved equation from the equation that should have been solved (directly calculatable)
Given the residual:
How can the error 𝜀^𝑛 be calculated from this?
By what factor do the memory requirements and computation time of the Gaussian elimination method increase if you refine the grid by a factor of 2 in each of the three spatial directions?
I. Memory requirements -> N2 -> ~(2*2*2)^2 ~64
II. Computation time -> N2 -> ~(2*2*2)^3 ~512
The iteration matrix G is often used to evaluate numerical solution algorithms. Name a sufficient criterion for the convergence of the method.
Necessary and sufficient for convergence is that the spectral radius of G (corresponds to the maximum eigenvalue) is smaller than 1.
What does the spectral radius ρ of the iteration matrix G tell us about its convergence behavior?
If the spectral radius of G is smaller than 1, the calculation converges. The smaller the spectral radius, the better.
Given the following spectral radii:
Which one would you choose and why?
a) The spectral radius is smaller than 1 & smallest for high X/Y values (fine grids) -> best convergence behaviour
What is meant by a coupled ILU solver? Name advantages and disadvantages.
Coupled Incomplete LU Factorization:
It solves a system of equations consisting if U-Impulse, V-Impulse, W-Impulse & Mass at the same time
More robust and faster convergence, i.e. less iterations
Higher storage requirements and higher computational effort per iteration compared to uncoupled solvers
What is a multigrid method? Explain the approach.
Observation:
Iterative solvers reduce those errors efficiently, that have a wavelength of the same order as the grid width!
Errors with small wavelengths are reduced quite quickly. If the wavelength is larger, numerous iterations are necessary.
Multi-grid method:
Convergence acceleration by utilization of multi-grids:
Adapt the grid to the wavelength of the error.
Early iterations on the fine grid and later iterations on a coarser grid
What is meant by restriction and prolongation in the context of multigrid methods?
How is coarsening done in the algebraic multigrid method?
Coarsening based on the coefficient-matrix:
-> The discrete equations (coefficients) for the coarse grid are obtained by summing up the fine grid equations (coefficients)
How is coarsening done in the geometric multigrid method?
Coarsening based on geometry and grid:
-> Groups the cells according to a geometric rule
Describe the approach of multigrid methods and explain the benefits.
Name two approaches to restriction.
Method:
Iterations on the fine grid
Calculation of the residuum
Transfer residuum to coarse grid (restriction)
Iterate correction equation on the coarse grid
Interpolate correction onto fine grid (prolongation)
Update solution on the fine grid
Repetition of the method until the required residuum is reached
Benefits:
Convergence acceleration
Approaches to restriction:
Algebraic Multi-Grid Method
Geometrical Multi-Grid Method
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