4A2 Computational Fluid Dynamics

Jie Li CUED 2020

• Computation + Fluid Dynamics

• Fluid Dynamics or Fluid Mechanics?

• Course Content

• Numerical Methods for Compressible Flows

Courtesy of Prof. Tucker, CUED

CFD

MEASUREMENTS THEORY

TRIAD (Available Tools)

• Making measurements

• Use of analytical solutions

• Use of Computer simulations (CFD- aerospace – 1960s)

• provide a significant amount of detail about a flow situation

• provide an effective means for the rapid evaluation of what-if design scenarios

• Geometry changes easy

• Safe for dangerous experiments

CFD Advantages

• Turbulence modelling a big problem

• Geometry handling and meshing can be time consuming

• Needs careful validation

CFD Problems

Course Structure

• Introduction

• Numerical Basics

• Writing a Basic Euler Solver

• Advanced concepts and Test Cases

• Total Variation Diminishing (TVD) Methods

• High Resolution Methods

• Riemann solver and shock capturing

Coursework Example: Subsonic Flow over a Bump

Mesh

Mach Contours

Course Information

Online Demonstration Sessions (Oct 15 - Dec 2)

▪ Mondays (2pm - 4pm)

▪ Tuesdays (2pm - 4pm)

▪ Wednesdays (2pm - 4pm)

▪ Thursdays (2pm - 4pm)

Documents (on Linux Dept. Teaching System)

Submission of 2 Reports

▪ ls /public/teach/4A2/

▪ cp -r /public/teach/4A2/Reading .

▪ cp -r /public/teach/4A2/2020_4A2 .

▪ Interim report: before 4pmThursday 12th Nov

▪ Final report: before 4pm Friday 11th Dec

Interim Report (due Nov 14)

• 4A2 Basic Euler Solve for Internal Flow

• F90 Programming Language

• More Explanation

Final Report (due Dec 13)

▪ 4A2 Euler Extensions

▪ More Explanation

• Test Cases

• Examples

Course Work (note on Moodle)

Access to Linux Teaching System

For Windows Platform, first install MobaXterm

https://mobaxterm.mobatek.net/download-home-edition.html

From Linux Platform

▪ ssh –X CRSid@gate.eng.cam.ac.uk

▪ ssh –X CRSid@ts-access

DATA Transfer▪ scp CRSid@gate.eng.cam.ac.uk:2020_4A2/SaveSrc.tar.gz .

▪ scp SaveSrc.tar.gz CRSid@gate.eng.cam.ac.uk:2020_4A2/.

Use of MobaXterm

Launch MobaXterm

Start local terminal

Oblique Shock & Prandtl-Meyer Expansion Wave(Compressible Flows)

high order method simulation

Shock Wave Reflection from a Wedge(Compressible Flows)

Ma = 1.7, wedge 25 degree

high order method simulation experiment (Prof. Takayama)

Diffraction of a Shock Wave By a Finite Wedge

Photographs by H. Schardin, in Oertel

Diffusion, Dissipation and High Order Method

(From LeVeque, CUP)

Diffusion, Dissipation and TVD Method

(From LeVeque, CUP)

TVD method

• Flow around NACA0012

Re = 1000

• Von Karman

Vortex Street

experiment (Shen et al.) high order method simulation

Incompressible Flows(Pressure based Solver)

Compressible Flows vs Incompressible Flows

Mach Number: U/a, a =: sound speed

Courant–Friedrichs–Lewy (CFL) condition:

Incompressible flow:

This course focuses on compressible flows

Explicit Methods, not suitable for incompressible flows

F/A-18, Sound barrierCourtesy of wiki

Hypersonic and High-Temperature Flows

• Ma > 5.0?

• High temperature

• Plasma flows

Sir Horace Lamb : ‘I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics and the other is the turbulent motion of fluids. About the former I am rather optimistic’

4A12 Module

Photograph by Werle, Courtesy of Delery

• aluminium wall of 3 meter thick

• 37 m high and 30 m in diameter

Space Power Facility (NASA, 1969)

Continuum Hypothesis vs Rarefied Gas Effect

Continuum Hypothesis vs Rarefied Gas Effect

• Mean free path: (100 nm at 1 atm)

• Characteristic Length:

• Knudson number:

• Rarefied gas flow:

Boltzmann Equation (Module 3A9):

Aspects of CFD

• Mesh Generators

• Structured/Unstructured mesh methods

• Numerical Methods (FDM, FVM, FEM…)

• Turbulence Modelling

• Compressible Flows (shock capturing)

• Incompressible Flows (pressure based methods)

• Eulerian/Lagrangian (moving/fixed methods)

• Adaptive Mesh Methods

• Parallel Computing

Journal of Computational Physics (Journal)

Conservation Equations of Compressible Flows

Cauchy Stress Tensor:

Stokes:

Fourier heat flux:

Mass:

Momentum (f body forces) :

Energy (Q heat sources):

Navier-Stokes Eqns for Compressible Flows

Polytropic gas:

Total Energy:

Ideal gas:

Stokes Viscous stress:

Fourier heat flux:

Constitutive Laws:

Boundary Conditions:

Mass:

Momentum:

Energy:

no-slip on wall, ….

DNS, LES and RANS

• Direct Numerical Simulations (DNS)

• Kolmogorov scale:

• 3D DNS requires mesh points :

• Large Eddy Simulations (LES)

• Reynolds Average Navier-Stokes Eqns (RANS)

Euler Equations

In 4A2:

Tensorial Product:

Boundary conditions: slip condition on wall, ….

4A2: Euler Equations in 2D

Speed of Sound:

Ideal gas:

Polytropic gas:

Energy:

Momentum:

Mass:

Conforming Boundary and Non-Conforming methods(no-slip condition on wall)

• Cut-cell methods

• Immersed boundary methods

• Flow passing a cylinder • Body fitting methods

Structured and Unstructured Meshes

• Structured : fixed neighbour relations, efficient

• Unstructured: variable neighbour relations, flexible

• Mixed type meshes

Delaunay Mesh Generation

Edge Flip:

Mesh Generators in Public Domain

• Triangle (Prof. J. Shewchuk)

• TetGen (Dr. Hang Si)

Lagrangian and Eulerian Approaches

Bouncing of Colliding Tetradecane Droplets

(Fixed or Moving Mesh Methods)

Adaptive Mesh Refinement (AMR)

AMRexperiment

Berger & Colella (1989). "Local adaptive mesh

refinement for shock hydrodynamics". J. Comput.

Phys. 82: 64–84

Moving Adaptive Mesh Techniques

• Conforming Moving Adaptive

Mesh Methods

• Adaptive Mesh Refinement

using quad-tree

Software Packages in Public Domain

• FENICS

• OpenFoam

• Gerris

• Fludity

• FreeFEM

(Oct-Tree)

Moore’s Law Moore's law: the number of transistors in a dense

integrated circuit doubles about every two years.

Parallel Computing • Memory Limitation & Computation Time

• Message Passing Interface (MPI)

• ParMetis: Mesh Partition

• Linear Solver: PETSc

Partitioned into 16 Parts

using ParMetis

• Summit(Oak Ridge National Lab, 2018)

• 4608 nodes

• 9,216 CPUs & 27,648 GPUs

• speed: 200 petaflops

Parallel Computation of 64 Rising Bubbles

Rising bubbles computed

on 16 cores by Shidi Yan

• Limit of Memory of a Single Machine

• Duration of computing time

• Partition of Mesh by ParMetis

• Incompressible Flows

• Pressure based method (implicit on pressure)

• Linear Solver PETSc

Euler Equations in 2D

Enthalpy:

Ideal gas:

Polytropic gas:

Energy:

Momentum:

Mass:

• Finite Element methods are based on weak form

of PDE, require a lot of maths (module 3D7)

Conservation Laws and Finite

Volume method

Writing the basic

solver Pt 1

Tom Hynes

Task at Hand

Initial Calls

program euler

call read_data

call inguess

call generate_grid

call check_grid

call crude_guess !flow_guess

call set_timestep

Main Loop Calls

do nstep = 1,nsteps

ro_start = ro

call set_others

call apply_bconds

call set_fluxes

call sum_fluxes

ro = ro_start + ro_inc

call smooth

call check_conv

end do

Read Data

Generate Grid

Generate Grid

Generate grid

Generate Grid

Generate Grid

Check Grid

Set Time Step

Set Time Step

Set Others

Apply Boundary Conditions

Set Fluxes

Sum Fluxes

del_prop(i,j) = (deltat/area(i,j))*(iflux(i,j) – iflux(i+1.j) +jflux(i,j)-

jflux(i,j+1))

Sum Fluxes

del_prop(i,j) = (deltat/area(i,j))*(iflux(i,j) – iflux(i+1.j) +jflux(i,j)-

jflux(i,j+1))

prop_inc(i,j) =

Smooth

Flow Guess

Flow Guess

Summary