MECH3780 - Computational Fluid Dynamics

What four equations make up the Navier Stokes equations

Continuity equation + (x,y,z) momentum equations

What terms make up the Navier Stokes equations

Variation + Divergence = Internal Source + Diffusion + External Source

Which behavior is dominant in a flow with a large RE and how does it affect the NS equations

Inertial flows are dominant, hence resulting in viscous terms (2nd order) cancelling out

Which behavior is dominant in a flow with a small RE and how does it affect the NS equations

Viscous flows are dominant, hence resulting in intertial terms (1st order) cancelling out

Why are Finite Difference schemes used?

To approximate derivatives and generate a set of algebraic equations that can be solved

How do you generate a discrete version of an equation

Substitute appropriate finite difference schemes and rearrange for appropriate variable

Controlling what is a crucial part of any solution for FD schemes

The error magnification for practical step sizes

What is truncation error

Truncation error is the error made by cutting off an infinite sum and approximating it by a finite sum. It is typically denoted as 𝑂( ∆𝑡)

What is up-winding, down-winding?

Where the information flows against and with the increasing node respectively

Difference between numerical diffusion and physical diffusion

Numerical diffusion is the artificial spreading of features due to truncation error of discretizing a solution. Smaller for grid spacing.

Physical diffusion is where molecular or turbulent spreading occurs in physical systems

What results in the convection-diffusion equation?

Combined convective and diffusive processes

How is the Laplace equation generated?

With the time-dependent term in the diffusion equation disappearing

What is more pronounced when discontinuities are present?

Differences in formulation and discretization

Where can PDEs be no longer applicable?

Where there is a formation of gradient catastrophes (where the gradient becomes infinite)

What are shocks?

Discontinuities which are governed by integral relations, not by differential relations. These are determined by conservation properties

Why do shocks arise?

Due to the nonlinear convection in fluids flow and are best handled with conservative schemes

What is the Finite Volume method

A numerical technique used to solve partial differential equations by dividing the computational domain into discrete control volumes or cells and approximating the fluxes of conserved quantities across the cell interfaces. Well-suited for conservation laws and provides a conservative and locally balanced approach to solving differential equations.

What are some disadvantages of FVM?

1. Difficult to develop in 3D

2. Requires all three levels of approximation: Interpolation, Differentiation, and Integration

3. More nodes required increases solution complexity

4. Boundary handling can become awkward

What are Dirichlet boundary conditions

Denoted as 𝜙, Dirichlet boundary conditions are a type of boundary which specifies the values of the unknown variables directly on the boundaries of the domain, effectively fixing or prescribing the solution at those locations.

What are the reasons for differences between results and 'reality'?

1. Modelling errors

2. Discretization errors

3. Iteration errors

What is a Direct Solution?

A directly computed solution from a problem using a finite number of steps. An example of a method includes Gaussian Elimination

What is an Iterative Solution

A method which repeatedly refines an initial guess to approach the desired solution, often used when direct solutions are computationally expensive or not feasible. Examples of methods includes Jacobi and Gauss-Seidel methods

What is the difference between the Jacobi and Gauss-Seidel methods?

The Gauss-Seidel method is a modification of the Jacobi method. The difference is that the GS method has a condition where the value of the variables are modified as soon as new values are evaluated. The Jacobi method is not modified until the next iteration

What is a residual?

Provides a measure of the solution error, rather than the absolute error. The residual represents the change to the solution of the equation. providing one value per matrix row.

Why is Pressure-Velocity coupling and how is it solved?

PVC is necessary to ensure that the computed pressure and velocity fields are consistent with each other. These arise from there being no equation for pressure, with the continuity equation representing a restriction on the computed velocity field. The matrix form can be arranged and solved using SIMPLE, PISO and other algorithms

What makes a transient flow more numerically stable?

The small ∆t in transient flows makes the diagonal terms large, resulting in more numerically stable

What is the difference between the SIMPLE and PISO algorithm?

SIMPILE is recommended for steady state flows, where there is no time derivative. SIMPLE is intended for less diagonally dominant, more numerically unstable solutions, where under-relaxation is required. PISO is recommended for transient, incompressible flows. Greatly reduces cost per iteration.

What are boundary conditions?

They are a set of defined parameters which mimic the fluid physics in a matter which can be solved numerically. The analysis of BC sensitivity is prudent and should be used to minimize computational cost

What is the recommended BC?

A velocity inlet and pressure outlet, which allows for rapid convergence and computational robustness

What can propagate disturbances against the direction of flow?

Sound waves if the speed is subsonic

Where should outlets be placed?

1. As far as possible from the specific region of interest

2. Avoided in regions of distinct geometrical change or wake regions featuring recirculation

3. Positioned at regions where the flow is approximately unidirectional

What two BC techniques can be used to reduce the size and required domain?

Symmetry (which is the same about a plane), and Periodicity (which is continuous across a plane)

What are the two processes of formulating and solving a discrete approximation?

1. Discretization of Equations - Numerical Method

2. Discretization of Domain - Mesh Generation

What is a mesh and how can it's quality be determined?

Collection of cells on which a flow is resolved. Quality can be assessed by the density of cells, discontinuities in cell sizing, skewness, cell type, boundary layer, static or dynamic mesh refinement through adaptation.

What is a structured mesh?

Families of grid lines with the property that members of a single family do not cross each other, and cross each member of the other families only once.

- Enhances computational performance and accuracy

- Time consuming to construct and often infeasible for complex geometries

What is a un-structured mesh?

Any grid that does not hold the property of a structured mesh. Provides significantly more flexibility.

- More computational overhead required

- Can introduce additional numerical errors

What is a hybrid mesh?

Combines structured mesh regions near boundaries and unstructured mesh where flow less important or geometry too complex.

What are measures of mesh quality?

1. Skewness

2. Smoothness

3. Aspect Ratio

What is turbulence?

Turbulence is irregular motion which in general makes its appearance in fluids, gaseous or liquid, when they flow past solid surfaces or even when neighboring streams of the same fluid flow past or over one another

What dominant environmental factor causes Turbulence

A high Reynold's Number RE

What are the properties of turbulence?

1. Chaotic, disorganized behavior

2. Sensitive to initial conditions

3. Large range of length and time scales

4. Enhances mixing and dissipation

5. 3D, time dependent, rotational

6. Intermittent behavior in both space and time

What is RANS?

Reynold's Averaged Navier Stokes model. Most common types of turbulence models are based on RANS, unsteady state. Uses additional equations to roughly account for turbulent affect. Generally adds viscosity where turbulence is expected.

What is the Kolmogorov scale?

The Kolmogorov scale refers to the smallest length scale in a turbulent flow where the effects of turbulence become dominant. It represents the size at which energy is transferred from large-scale turbulent motions to small-scale fluctuations.

What is DNS?

Direct Numerical Simulation. Resolves all turbulence, though need to consider Kolmogorov Scale. great for simple (and small) flows and scenarios, but computational intensive, models both small and large Eddies.

What is LES

Large Eddy Simulation. LES sits between DNS and RANS in both Rigor and Computational requirement. Small eddies are removed and modelled with a sub grid-scale, with the simulation being transient. Great for large scenarios

What are the examples of when LES is to be used over RANS?

1. Flows with large separation

2. Bluff-body flow

3. Transitional flow

What is the Boussinesq Assumption?

Assumes the density of a fluid remains constant except for small changes due to temperature variations. Simplifies governing equations by treating the fluid as incompressible and allows for the application of simplified equations such as NS

What are the advantages and disadvantages of the Prandtl mixing length model?

Advantages: Easy to implement, computational cheap, reasonable prediction of thin shear and boundary layers, well established.

Disadvantages: Incapable of describing flows not near equilibrium, only calculates mean flow properties and shear stress

What are the advantages and disadvantages of the Spalart-Allamaras Model?

Advantages: Cheap, only one additional equation, good performance for external flows (aircraft design), good performance in boundary layers with adverse pressure gradients.

Disadvantages: Issues with free vortex flows, lacks sensitivity to transport processes in rapidly varying flows, not particularly good if very large separation

List the different types of fluids and how they are identified on a stress strain-rate graph

1. Newtonian (Start from origin and are linear in nature)

2. Shear-thickening (start from origin and have an exponential increase in stress)

3. Shear-thinning (start from origin and have an exponential increase in strain-rate)

4. Casson Fluid (Start with a stress tau>0 and have an exponential increase in strain rate)

5. Bingham Plastic (Start with a stress tau>0 and is linear in nature)

List examples of the different types of fluid

1. Newtonian (water, air milk, alcohol)

2. Shear-thickening (dense suspensions, wet cement)

3. Shear-thinning (paint, whipped cream, tomato sauce)

4. Casson Fluid (Blood, yoghurt)

5. Bingham Plastic ( toothpaste, clay, drilling mud)

Which fluids exhibit a yield stress?

Both Bingham plastic and Casson Fluids due to having a shear stress when the strain rate is 0

What is a power-law fluid?

Non-Newtonian fluid which exhibits a non-linear relationship. Shear-thickening/thinning and Casson fluids are power law, denoted n.

n<1 is shear-thinning

n=1 is Newtonian

n>1 is shear-thickening

What is the Carreau Model?

Constitutive equation used to describe the Rheological (flow and deformation) behavior of viscoelastic fluids

What is conduction, convection, and radiation?

Conduction: Diffusivity of heat due to temperature gradients

Convection: Heat is carried away by fluid motion. Two types of convection: Forced, and Natural convection

Radiation: Energy transfer without requirement of a medium

What are the three common forms of the energy equation?

1. Total energy Equation

2. Enthalpy Equation

3. Thermal/Internal Energy equation

What is the Thermal Boundary layer?

Depends on the fluid properties characterized by the Prandtl number

High Prandtl (thermal sublayer < momentum sublayer). Low Prandtl (thermal sublayer > momentum sublayer)

What is Conjugate heat transfer

CHT is the ability to calculate conduction of heat through solids coupled with convective heat transfer in a fluid. Requires both fluid and solid zones to be meshed and resolved

What is a multiphase flow?

Simultaneous transport of matter with two or more phases, or matter with varying properties.

These fluids are mixed at the molecular level or macroscopic level.

Challenges of Computational Multi-Fluid Dynamics

1. Inherently multiscale

2. No universal approach exists

3. Complex topology evolution

4. Additional physics to be captured

4. High numerical gradients

Types of multiphase flows

1. Separated multiphase flow

2. Dispersed multiphase flow

3. Mixed phases

Modelling approaches to multiphase flows

1. Empirical correlations

2. Lagrangian

3. Algebraic slip model

4. Eulerian model

5. Discrete element method

6. Volume of Fluid

What is Discrete Phase Modelling?

Trajectory of particles/droplets/bubbles computed in a Lagrangian frame.

What is Eulerian model?

Provides larger volume factions of dispersed phases, with the phases being fluid, or solid

What is Volume of Fluid model?

Uses immiscible condition that allows the combination of momentum equations.