cfdprocess

CFD 9 – 1 David Apsley

9. THE CFD PROCESS SPRING 2012 9.1 Introduction 9.2 The computational mesh 9.3 Boundary conditions 9.4 Flow visualisation Examples 9.1 Introduction 9.1.1 Stages of a CFD Analysis A complete CFD analysis consists of: • pre-processing; • solving; • post-processing. This course has focused on the “solving” process, but this is of little use without pre-processing and post-processing facilities. Commercial CFD vendors supplement their flow solvers with grid-generation and flow-visualisation tools, as well as graphical user interfaces (GUIs) to simplify the setting-up of a CFD analysis. Pre-Processing The pre-processing stage consists of: – determining the equations to be solved; – specifying the boundary conditions; – generating a computational mesh or grid. It depends upon: – the desired outputs of the simulation (e.g. force coefficients, heat transfer, ...); – the capabilities of the solver. Solving In commercial CFD packages the solver is often operated as a “black box”. Nevertheless, user intervention is necessary – to set under-relaxation factors and input parameters, for example – whilst an understanding of discretisation methods and internal data structures is required in order to supply mesh data in an appropriate form and to analyse output. Post-Processing The raw output of the solver is a huge set of numbers corresponding to the values of each field variable (u, v, w, p, …) at each point of the mesh. This must be reduced to some meaningful subset and/or manipulated further to obtain the desired predictive quantities. For example, a subset of surface pressures and cell-face areas is required to compute a drag coefficient. Commercial packages routinely provide:

– plotting tools to visualise the flow; – analysis tools to extract and manipulate data.


9.1.2 Commercial CFD The table below lists some of the more popular commercial CFD packages. Developer/distributor Code(s) Web address (liable to change!) CD adapco STAR CD

STAR CCM+ http://www.cd-adapco.com/

Ansys FLUENT CFX

http://www.ansys.com/Products/

Flow Science FLOW3D http://www.flow3d.com/ EXA PowerFLOW http://www.exa.com/ CHAM PHOENIX http://www.cham.co.uk/ A popular non-commercial CFD solver is OpenFOAM (http://www.openfoam.com/) An excellent web portal for all things CFD is http://www.cfd-online.com/. 9.2 The Computational Mesh 9.2.1 Mesh Structure The purpose of the mesh generator is to decompose the flow domain into control volumes. The primary outputs are: • cell vertices; • connectivity information. Precisely where the nodes are relative to the vertices depends on whether the solver uses, for example, cell-centred or cell-vertex storage. Further complexity is introduced if a staggered velocity grid is employed.

The shapes of control volumes depend on the capabilities of the solver. Structured-grid codes use quadrilaterals in 2-d and hexahedra in 3-d flows. Unstructured-grid solvers often use triangles (2-d) or tetrahedra (3-d), but newer codes can use arbitrary polyhedra.

hexahedron

tetrahedron

cell-centred storage

cell-vertex storage

u p

v

staggered velocity mesh


In all cases it is necessary to specify connectivity: that is, which cells are adjacent to each other, and which face they share. For structured grids, with (i,j,k) numbering this is straightforward, but for unstructured grids quite complicated data structures must be set up to store connectivity information. 9.2.2 Areas, Volumes and Cell-Averaged Derivatives (*** Advanced ***) In continuum mechanics, conservation laws take the form:

rate of change + net flux = source To calculate fluxes requires the vector areas of cell faces; e.g. Au •=fluxmass To find the total amount of some property in a cell requires its volume; e.g. φ= Vamount Areas and volumes are easy to evaluate in Cartesian meshes, but general-purpose CFD, which employs arbitrary polyhedral meshes, requires more sophisticated geometrical techniques. Areas Triangles. The vector area of a triangle with side vectors s1 and s2 is

2121 ssA ∧=

The orientation depends on the order of vectors in the cross product. Quadrilaterals 4 points do not, in general, lie in a plane. However, since the sum of the outward vector areas from any closed surface is zero, i.e.

0d =⌡

⌠

∂V

A or 0=∑faces

fA

any surface spanning the same set of points has the same vector area. By adding vector areas of, e.g., triangles 123 and 134, the vector area of any surface spanned by these points and bounded by the side vectors is found (see Examples) to be half the cross product of the diagonals:

)()( 241321

241321 rrrrddA −∧−=∧=

(Again, the order of points determines the orientation of the area vector). General polygons The vector area of an arbitrary polygonal face may be found by breaking it up into triangles and summing the individual vector areas. Again, the vector area is independent of how it is broken up.

s1

s2

A

A

A

r3

4r

1r 2r


Volumes

If ),,( zyx≡r is the position vector, then 3=∂∂+

∂∂+

∂∂=•∇

z

z

y

y

x

xr . Hence, integrating over

an arbitrary control volume and using the divergence theorem gives for the volume of a cell:

⌡

⌠ •=∂V

V Ar d3

1

If the cell has plane faces this can be evaluated as

∑ •=faces

ffV Ar3

1

where rf is any convenient position vector on a face and Af is the face vector area, since, for any other vector r in that face,

44344210

)(

=

•−+•=• fffff ArrArAr

The last term vanishes because r – rf is perpendicular to Af for any point on a plane face. If the cell faces are not planar, then the volume depends on how these faces are broken down into triangles. Typically the area of a particular face can be regarded as an assemblage of triangular elements connecting the vertices of that face with a central reference point formed, e.g.

∑=vertices

if Nrr

1

Examples for the commonest shapes follow. Tetrahedra

vectors s1, s2, s3 The volume of a tetrahedron formed from side (taken in a right-handed sense) is

32161 sss ∧•=V

Hexahedra Volumes of arbitrary polyhedral cells are:

∑ •=faces

ffV Ar3

1

For hexahedra, on each face the reference point and vector area are: )( 43214

1 rrrrr +++=f

)()( 423121 rrrrA −∧−=f

To obtain an outward-directed face vector, the points with position vectors r1, r2, r3, r4 should be in clockwise order when viewed along the outward normal (right-hand screw). This is a generalisation to arbitrary hexahedra of the result for Cartesian control volumes.

rf

s3

1s

2s


2-d Cases In 2-d cases one can formally consider cells to be of unit depth. The “volume” of the cell is then numerically equal to its planar area. The side “area” vectors are most easily obtained from their Cartesian projections:

−=0

d

d

d x

y

A

To obtain an outward-directed dA, the edge increments (dx,dy,0) must be taken anticlockwise around the cell. Volume-Averaged Derivatives By applying the divergence theorem to φex, where ex is the unit vector in the x direction, the volume-averaged derivative of a scalar field φ is

⌡

⌠ φ=⌡⌠

∂φ∂=

∂φ∂

∂Vx

V

AV

VxVx

d1

d1

or

∑φ=φ∇faces

ffVA

1

Thus, for arbitrarily-shaped cells, average derivatives may be derived from the values of φ on the cell faces, together with the components of the face area vectors and the cell volume. e.g. for hexahedra:

)(1

/

/

/

ttbbnnsseewwVz

y

x

AAAAAA φ+φ+φ+φ+φ+φ=∂φ∂∂φ∂∂φ∂

The values on cell faces, φw, φe etc. can be obtained by interpolation from the nodes on either side.

φe

φn

wφφs

Ae

An

Aw

As

s

Ad

ddy

dx

V


9.2.3 Classification of Grid Types Grids can be Cartesian or curvilinear (usually body-fitting). In the former, grid lines are always parallel to the coordinate axes. In the latter, coordinate surfaces are curved to fit boundaries. There is an alternative division into orthogonal and non-orthogonal grids. In orthogonal grids (for example, Cartesian or polar meshes) all grid lines cross at 90°. Some flows can be treated as axisymmetric, and in these cases, the flow equations can be expressed in terms of polar coordinates (r, ), rather than Cartesian coordinates (x,y), with minor modifications. Structured grids are those whose control volumes can be indexed by (i,j,k) for i = 1,..., ni, j = 1,.., nj, k = 1,..., nk, or by sets of such blocks (multi-block structured grids – see below). Each structured block of control volumes, even if curvilinear, can be distorted by a coordinate transformation into a cube (or square in two dimensions). Multi-block structured meshes can be used for many practical flow configurations. Unstructured meshes can accommodate completely arbitrary geometries. However, there are significant penalties to be paid for this flexibility, both in terms of the connectivity data structures and solution algorithms. Grid generators and plotting routines for such meshes are also very complex.

unstructured Cartesian mesh

unstructured triangular mesh

single-block structured cartesian mesh

single-block structured curvilinear mesh


9.2.4 Fitting Complex Boundaries With Structured Grids Blocking Out Cells The range of flows which can be computed in a rectangular domain is rather limited. Nevertheless, a number of significant bluff-body flows can be computed using a single-block Cartesian mesh by the process of blocking out cells. Solid-surface boundary conditions are applied to cell faces abutting the blocked-out region, whilst values of velocity and other flow variables are forced to zero by a modification of the source term for those cells. In the notation of Section 4, where the scalar-transport equations for a single cell are discretised as

PPPFFPP sbaa φ+=φ−φ ∑

the source terms are simply re-set: ,0 −→→ PP sb

where is a large number (e.g. 1030). Rearranging for φP, this ensures that the computational variable φP is effectively forced to zero in these cells. However, the computer still stores and carries out operations for these points, so that it is essentially performing a lot of redundant work. An alternative approach is to fit several structured grid blocks around the bluff body. Multi-block grids will be discussed further below. Volume-of-Fluid Approach The numerical simplicity and solver efficiency associated with Cartesian grids mean that some practitioners still attempt to retain this grid geometry even for complex curved boundaries: both for solid walls and free surfaces. In this volume-of-fluid approach the fraction f of the cell filled with fluid is stored: 0 outside the fluid domain, 1 within the interior of the fluid and 0 < f < 1 for cells which are cut by a boundary. At solid boundaries f is determined by the surface contour. At free surfaces, f emerges as part of the solution procedure.

mesh for 2-d rib with blocked-out cells

f=0

f=1

0<f<1

f=0


Body-Fitted Grids The majority of general-purpose codes employ body-fitted (curvilinear) grids. The mesh lines/coordinate surfaces are distorted so as to fit snugly around curved boundaries. Accuracy in turbulent-flow calculations demands a high density of grid cells close to solid surfaces and the use of body-fitted meshes means that the grid need only be refined in the direction normal to the surface, with consequent saving of computer resources. However, the use of body-fitted grids has important consequences. • It is necessary to store detailed geometric components for each control volume; for

example, in our research code STREAM we need to store (x,y,z) components of the cell-face-area vector for “east”, “north” and “top” faces of each cell, plus the volume of the cell itself – a total of 10 arrays.

• Unless the mesh happens to be orthogonal, the diffusive flux

through the east face (say) is no longer given exactly by

AAn PE

PE φ−φ=

∂φ∂

because the discretised derivative of φ normal to the face involves cross-derivative terms parallel to the cell face and nodes other than P and E. The extra off-diagonal diffusion terms are typically transferred to the source term.

• A similar alignment problem occurs with the advection terms because, in general, interpolated values of all three velocity components are needed to evaluate the mass flux through a single face. This necessitates approximations in the pressure-correction equation that can slow down the solution algorithm.

Multi-Block Structured Grids In multi-block structured grids the domain is decomposed into a small number of regions, in each of which the mesh is structured (i.e. cells can be indexed by (i,j,k)). A common arrangement (and that assumed by our own code STREAM), is that grid lines match at the interface between two blocks, so that there are cell vertices that are common to two blocks. However, some solvers do allow overlapping blocks (chimera grids) or block boundaries where cell vertices do not align. Interpolation is then needed at the boundaries of blocks.

2 3 451

2-d rib with multi-block mesh

PE η=const.

const.ξ =

u

v


In the usual arrangement, with both cell vertices matching at block boundaries and blocks meeting “whole-face to whole-face” the solver starts by adding additional lines of cells from one block to those of the adjacent, so that, in discretising, accuracy is not compromised. On each iteration of a scalar transport equation the discretised equations would be solved implicitly within each block, with values from the adjacent block providing boundary conditions. At the end of each iteration, values in overlap cells would be explicitly updated using data from the interior of the adjacent block.

Multiple blocks are employed to maintain a structured grid configuration around complex boundaries. There are no hard and fast rules, but it is generally desirable to avoid sharp changes in grid direction (which lead to lower

accuracy) in important and rapidly changing regions of the flow, such as near solid boundaries. One should also strive to minimise the non-orthogonality of the grid. 9.2.5 Discretisation on Curvilinear Meshes Non-Cartesian coordinate systems are called curvilinear. The coordinates are denoted (, , ) or ( i) and the direction of the coordinate lines varies with location. Coordinate systems in which coordinate lines cross at right angles are termed orthogonal. Cartesian systems are obviously in this category, but other commonly-used orthogonal systems include cylindrical and spherical polar coordinates. However, most curvilinear systems are non-orthogonal. The direction normal to one face is not necessarily along a coordinate line, and hence the diffusive flux, which depends on ∂φ/∂n, cannot be approximated solely in terms of the nodes either side of the face. For example, in 2 dimensions, if the east face ( = constant) of a control volume coincides with x = constant, the normal derivative is

xxx ∂

∂∂φ∂+

∂∂

∂φ∂=

∂φ∂

The first derivative, ∂φ/∂ , can be discretised as /)PE φ−(φ and

treated implicitly, but the derivative parallel to the cell face, ∂φ/∂ , depends on the values at other nodes; it has to be transferred to the source side of the equation and treated explicitly. The coordinate derivatives ∂ /∂x, ∂ /∂y, can be calculated analytically or (my preference) determined from the cell volume and the cell-face-area

chimera grid

2 3 458

761

PE η=const.

const.ξ =


vectors; e.g.

)�

(),,( AVzyx

→∇≡∂∂

∂∂

∂∂

where V is volume and )�

(A is the area vector normal to face = constant (in the direction of increasing ). Curvilinear coordinates can be taken as the node indices, in which case = 1. Non-orthogonal meshes are more computationally-intensive, but their additional geometric flexibility makes them desirable in general-purpose solvers. 9.2.6 Disposition of Grid Cells Only two points are necessary to resolve a straight-line profile, but the more curved the profile the more points are needed to resolve it accurately. In general, more points are needed in rapidly-changing regions of the flow, such as: – solid boundaries; – separation, reattachment and impingement points; – flow discontinuities; e.g. shocks, hydraulic jumps. Simulations must demonstrate grid-independence, i.e. that a finer-resolution grid would not significantly modify the solution. This generally requires a sequence of calculations on successively finer grids. Boundary conditions for turbulence modelling do impose some limitations on the cell size near walls. Low-Reynolds-number models (resolving the near-wall viscous sublayer) generally require that y+ < 1 for the near-wall node, whilst high-Reynolds-number models relying on wall functions require (in principle) the near-wall node to lie in the log-law region, ideally 30 < y+ < 150. 9.2.7 Multiple Grids Multiple grids – combining cells so that there are 1, 1/2, 1/4, 1/8, ... times the number of cells in some basic fine grid – are used deliberately in so-called multigrid methods. These calculate the solution on alternately coarser and finer grids. The idea is that the solution is obtained quickly on the coarsest grid where the number of cells is small and changes propagate rapidly across the domain. The solution is then refined locally on the finest grid to obtain the most accurate solution. If two levels of grid are used then a process known as Richardson extrapolation may be used to both estimate the error and, possibly, refine the solution. If the basic advection-diffusion discretisation is known to be of order n and the exact (but unknown) solution of some property φ is denoted φ* , then one would expect the error φ – φ* to be proportional to n, where is the mesh spacing. Thus, for solutions φ� and φ2

� respectively on two grids with mesh spacing and 2 ,

n

n

C

C

)2(*�2

*�+φ=φ+φ=φ

Two equations give us two equations for two unknowns, φ* and C, which we can solve to get a better estimate of the solution:


12

2 �2

�*

−φ−φ

=φn

n

and the error in the fine-grid solution:

12

��2

−φ−φ

=n

nC

Unstructured-grid methods offer the possibility of local grid refinement: that is, adding more cells in regions where the error is estimated to be high. 9.3 Boundary Conditions There are a number of common boundary types. INLET • Inflow transported variables specified on the boundary, either by a predefined profile or

(my preference) by doing an initial 1-d, fully-developed-flow calculation. • Stagnation (or reservoir) total pressure and total temperature (in compressible flow) or total head (in

incompressible flow) fixed; usual inflow condition for compressible flow. OUTLET • Outflow zero normal gradient (∂φ/∂n = 0) for all variables. • Pressure as for outflow, except fixed value of pressure; usual outlet condition in

compressible flow if the exit is subsonic. • Radiation (or convection) prevent wave-like motions from reflecting at outflow boundaries by solving a

simplified first-order wave equation with outward-directed wave velocity. WALL • Non-slip wall the default case for solid boundaries (zero velocity relative to the wall; wall stress

computed by viscous-stress or wall-function expressions). • Slip wall only the velocity component normal to the wall vanishes; used if it is not

necessary to resolve a thin boundary layer on an unimportant wall boundary. OTHER • symmetry plane ∂φ/∂n = 0, except for the velocity component normal to the boundary, which is

zero; used where there is a geometric plane of symmetry, but also as a far-field boundary condition, because it ensures that there is no flow through, nor viscous stresses on, the boundary.

• periodic used in repeating flow; e.g. rotating machinery, regular arrays.


9.4 Flow Visualisation CFD has a reputation for producing colourful output and, whilst some of it is promotional, the ability to display results effectively may be an invaluable design tool. 9.4.1 Available Packages Visualisation tools are often packaged with commercial CFD products. However, many excellent stand-alone applications or libraries are available. Some of the more popular in CFD are listed below. Most have versions for various platforms, including Windows and linux. Some are aimed predominantly at the CFD user, but others are general-purpose visualisation tools. Some are even free! • TECPLOT (http://www.tecplot.com) • EnSight ( http://www.ensight.com) • ParaView (http://www.paraview.org) • AVS (http://www.avs.com) • Iris Explorer (http://www.nag.co.uk/visualisation_graphics.asp) • Dislin (http://www.mps.mpg.de/dislin) • Gnuplot (http://www.gnuplot.info) 9.4.2 Types of Plot x-y plots These are simple, two-dimensional graphs. They can be drawn by hand or by many plotting packages. They are the most precise and quantitative way to present numerical data and, since laboratory data is often gathered by straight-line traverses, they are a popular way of making a direct comparison between experimental and numerical data. Logarithmic scales also allow the identification of important effects occurring at very small scales, particularly near solid boundaries. They are widely used for line profiles of flow variables (particularly velocity and stresses) and for plots of surface quantities such as pressure or skin-

friction coefficients. One way of visualising the development of the flow is to use several successive profiles.


Line Contour Plots A contour line (“isoline”) is a line along which some property is constant. The equivalent in 3 dimensions is an “isosurface”. Any field variable may be contoured. In contrast to line graphs, contour plots give a global view of the flow field, but are less useful for reading off precise numerical values. If the domain is linearly scaled then detail occurring in small regions is often obscured. The actual numerical values of the isolines are sometimes less important than their overall disposition. If contour intervals are the same then clustering of lines indicates rapid changes in flow quantities. This is particularly useful in locating shocks and discontinuities. Shaded Contour Plots WARNING: colour plots are expensive to print or photocopy! This proviso apart, colour is an excellent medium for conveying information and good for on-screen and presentational analysis of data. Simple packages flood the region between isolines with a fixed colour for that interval. The most advanced packages allow a pixel-by-pixel gradation of colour between values specified at the cell vertices, together with lighting and other special effects such as translucency. Grey-scale shading is an option if plots are to be reproduced in black and white. Vector Plots Vector plots display vector quantities (usually velocity; occasionally stress) with arrows whose orientation indicates direction and whose size (and sometimes colour) indicates magnitude. They are a popular and informative means of illustrating the flow field in two dimensions, although if grid densities are high then either interpolation to a uniform grid or a reduced set of output positions are necessary to prevent the number density of arrows blackening the plot. There can sometimes be


problems when selecting scales for the arrows when large velocity differences are present, especially in important areas of recirculation where the mean flow speed is low. In three dimensions, vector plots can be deceptive because of the angle from which they are viewed. Streamline Plots Streamlines are parallel to the mean velocity vector. They can generally be obtained by integration:

ux =td

d

(the only option in three dimensions). In 2-d incompressible flow a more accurate method is to contour the streamfunction . This function is defined (and calculated) by fixing

arbitrarily at one point and then setting the change in between two points as the volume flow rate (per unit depth) across any curve joining them:

⌡

⌠ •=−2

112 dsnu

(The sense used here is clockwise about the start point, although the opposite sign convention is equally valid). is well-defined because the flow rate across any path connecting two points must be the same or mass would accumulate between them. Contouring produces streamlines because a curve of constant is one across which there is no flow. For a short path making orthogonal increments dx and dy then xvyu ddd −= and hence the velocity components are related to by

y

u∂∂= ,

xv

∂∂−=

Computationally, is stored at cell vertices. This is convenient because the flow rates across the cell edges are already stored as part of the calculation. In the Cartesian cell shown: yvs12 −=

xue23 +=

If isolines are at equal increments in , then clustering of lines corresponds to high velocities, whilst regions where the streamlines are further apart signify low velocities. However, as with vector plots, this has the effect of making it difficult to visualise the flow pattern in low-velocity regions such as recirculation zones and a smaller increment in is needed here. Particle paths can also be traced along solid surfaces using the wall stress vector. This often reveals important features connected with separation/reattachment/impingement on 3-d surfaces, which defines the basic flow topology.

vs

ue

ψ3

ψ2ψ1

1

2

1

2

dy

dxv

u


Mesh Plots The computational mesh is usually visualised by plotting the edges of control volumes – for one coordinate surface only in 3 dimensions. It can be very difficult indeed to visualise fully-unstructured 3-d meshes, and usually only the surface mesh and/or its projection onto a plane section is portrayed.

Composite Plots In 3 dimensions it is common to combine plots of the above types, emphasising the behaviour of several important quantities in one plot. In this way, some sense can be made of the highly complex flows now being studied with CFD.

Finally, just to put things in perspective ...

“On two occasions I have been asked [by members of Parliament], ‘Pray, Mr. Babbage, if you put into the machine wrong figures, will the

right answers come out’. I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question.”

Charles Babbage

(Robert Peel’s government eventually pulled the plug on Babbage’s research and he was unable to build his “analytical engine”. Ever since then the British Government has preferred

to fight wars on the other side of the world rather than funding scientific research!)


Examples Q1. (a) Explain what is meant, in the context of computational meshes, by: (i) structured; (ii) multi-block structured; (iii) unstructured; (iv) chimera. (b) Define the following terms when applied to structured meshes: (i) Cartesian (ii) curvilinear (iii) orthogonal Q2. Suppose that your CFD solver requires multi-block, structured, curvilinear grids, where grid nodes must match at block interfaces. Sketch suitable grid topologies that might be used to compute the following flow configurations. Use symmetry where appropriate and “body-fitting” meshes, not “blocked-out” regions. (a) Harbour configuration (treat as 2-d) (b) Circular to square-section pipe expansion (c) Outfall (treat as 2-d) Q3. (Computational Hydraulics Exam, May 2011 – part) (a) For wind-loading calculations a CFD calculation of

airflow is to be performed about the building complex shown in the figure. Sketch a suitable computational domain indicating the specific types of boundary condition that are applied at each boundary of the fluid domain. For each boundary type summarise the mathematical conditions that are applied to each flow variable (velocity, pressure, turbulent scalar).

(b) Define the drag coefficient for a building and explain how it would be calculated from

the raw data obtained in a CFD simulation.


Q4. (a) Two adjacent cells in a 2-dimensional Cartesian mesh are shown below, along with

the cell dimensions and some of the velocity components (in m s–1) normal to cell faces. The value of the stream function at the bottom left corner is A = 0. Find the value of the stream function at the other vertices B – F. (You may use either sign convention for the stream function).

A

D F

C

12

3

5

0.3 m 0.2 m

0.1 m

E

B

3

(b) Sketch the pattern of streamlines across the two cells in part (a). Q5. (Computational Hydraulics Exam, June 2009 – part) The figure below shows a quadrilateral cell, together with the coordinates of its vertices and the velocity components on each face. If the value of the stream function at the bottom left corner is = 0, find: (a) the values of at the other vertices; (b) the unknown velocity component vn. Q6. (Computational Hydraulics Exam, May 2010 – part) A numerical scheme known to be second-order accurate is used to calculate a steady-state solution on two regular Cartesian meshes A and B where the finer mesh A has half the grid spacing of mesh B. The values of the solution φ at a particular point are found to be 0.74 using mesh A and 0.78 using mesh B. Use Richardson extrapolation to: (a) estimate an improved value of the solution at this point; (b) estimate the error at this point using the mesh-A solution.

velocity face u v w 3 –3 e 3 1 s 0 2 n 1 vn

(4,5)

(3,1)(1,1)

(1,3)

s

e

n

w


Q7. Show that the vector area of a (possibly non-planar) quadrilateral is half the cross product of its diagonals; i.e. )()( 24132

124132

1 rrrrddA −∧−=∧=

Q8. (a) (*** Advanced ***) Derive the following formulae for the volume of a cell and the

cell-averaged derivative of a scalar field:

(i) ⌡⌠ •=

∂V

V Ar d3

1;

(ii) ⌡

⌠ φ=∂φ∂

∂VxA

Vxd

1;

where r is a position vector and dA a small element of (outward-directed) area on the closed surface V.

A tetrahedral cell has vertices at A(2, –1, 0), B(0, 1, 0), C(2, 1, 1) and D(0, –1, 1). (b) Find the outward vector areas of all faces. Check that they sum to zero. (c) Find the volume of the cell. (d) If the values of φ at the centroids of the faces (indicated by their vertices) are φBCD = 5, φACD = 3, φABD = 4, φABC = 2,

find the volume-averaged derivatives x∂φ∂

, y∂φ∂

, z∂φ∂

Q9. (MSc Exam, April 2005 – part) The figure shows part of a non-Cartesian 2-d mesh. A single quadrilateral cell is highlighted and the coordinates of the corners marked. The values of a variable φ at the cell-centre nodes (labelled geographically) are: φP = 2, φE = 5, φW = 0, φN = 3, φS = 1. Find: (a) the area of the cell;

(b) the cell-averaged derivatives x∂φ∂

and y∂φ∂

,

assuming that cell-face values are the average of those at the nodes either side of that cell face.

Q10. One quadrilateral face of a hexahedral cell in a finite-volume mesh has vertices at (2,0,1), (2,2,–1), (0,3,1), (–1,0,2) (a) Find the vector area of this face (in either direction). (b) Determine whether or not the vertices are coplanar.

P

N

E

S

W(3,3)

(0,4)

(2,0)

(-2,2)


Q11. (MSc Exam, May 2008) In a 2-dimensional unstructured mesh, one cell has the form of a pentagon. The coordinates of the vertices are as shown in the figure, whilst the average values of a scalar φ on edges 1 – 5 are: φ1 = –7, φ2 = 8, φ3 = –2, φ4 = 5, φ5 = 0 Find: (a) the area of the pentagon;

(b) the cell-averaged derivatives x∂φ∂

and y∂φ∂

.

Q12. (*** Advanced ***) (MSc Exam, June 2009) (a) Give a physical definition of the stream function in a 2-d incompressible flow, and

show how it is related to the velocity components. (b) Show that if the flow is irrotational then satisfies Laplace’s equation. If the flow is

not irrotational how is ∇2 related to the vorticity? The figure below and the accompanying table show a 2-d quadrilateral cell, with coordinates at the vertices and velocity components on the cell edges. The flow is incompressible. All quantities are non-dimensional. (c) Assuming that the stream function at the bottom left (SW) corner is 0, calculate the

stream function at the other vertices. (d) Find the y-velocity component vn, whose value is not given in the table.

(e) By calculating ⌡⌠ • su d for the quadrilateral cell estimate the cell vorticity.

velocity face u v w 3 2 e 3 –1 s 0 2 n 1 vn

(2,-4)

(5,1)

(1,3)

(-3,0)

(-2,-3)

1

23

4

5

(1,1)

en

ws

x

y

(3,1)

(4,4)

(0,3)


Q13. (MSc Exam, June 2010 – part) (a) The vertices of a particular tetrahedral cell in a

finite-volume mesh are (0,0,0), (4,0,0), (1,4,0), (1,2,4) Find: (i) the outward vector area of each face; (ii) the volume of the cell. (b) For the cell in part (a) a scalar φ has value 6 on the largest face, 2 on the smallest face

and 3 on the other two faces. Find the cell-averaged derivatives x∂φ∂

, y∂φ∂

, z∂φ∂

.

(c) For the cell in part (a) determine whether the point (1,2,1) lies inside, outside or on

the boundary of the cell. (You must justify your answer.) Q14. (*** Advanced ***) (MSc Exam, May 2011 – part) A quadrilateral cell in a 2-d finite-volume mesh is shown in the figure below. The coordinates of the vertices are marked in the figure and the velocities on the edges are given in the adjacent table. (a) Find the area A of the quadrilateral.

(b) Find the cell-averaged derivatives x

u

∂∂

, y

u

∂∂

, x

v

∂∂

, y

v

∂∂

.

(c) By calculating the line integral from the velocity and geometric data, confirm that

both the cell-averaged velocity derivatives and the discrete form of Stokes’ law,

⌡⌠ •=

∂Az A

su d1

,

give the same value for the cell-averaged vorticity, z . (Note: this is, in fact, true in

general)

(0,-1)

(2.5,2)

(-1,3)

(-2,0)e

n

w

sx

y

edge u v e 10 12 n 13 6 w 9 -6 s 1 0

x y

z

O

A(4,0,0)

B(1,4,0)

C(1,2,4)

cfdprocess

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