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2. FLUID-FLOW EQUATIONS SPRING 2012
2.1 Introduction
2.2 Conservative form of the flow equations
2.3 Non-conservative form of the flow equations
2.4 Non-dimensionalisation
Summary
Examples
2.1 Introduction
Fluid dynamics is governed by conservation equations for:
- mass;
- momentum;
- energy;
- (for a non-homogenous fluid) other constituents.
Equations for these can be expressed mathematically in many ways, notably as
- integral (i.e. control-volume ) equations;
- differential equations.
This course will focus on the integral (control-volume) approach because it is more directly
related to the physical world and forms the basis of the finite-volume method. However, the
equivalent differential equations (of which there are several forms) are often easier to write
down, manipulate and, in some cases, solve analytically.
Although there are different fluid-flow variables, most of them satisfy a single generic
equation: the scalar-transport or advection-diffusion equation.
The rate of change of some quantity within an arbitrary control volume is determined by:
- the net rate of transport across the bounding surface (“ flux ”);
- the net rate of production within that control volume (“ source ”).
inside V outof boundary inside V
RATE OFCHANGE FLUX SOURCE (1)
The flux across the bounding surface can be divided into:
1 : transport with the flow;
- diffusion : net transport by molecular or turbulent fluctuations.
inside V throughboundary inside V
RATE OFCHANGE ADVECTION DIFFUSION SOURCE (2)
The finite-volume method is a natural discretisation of this.
(^1) Some authors – but not this one – prefer the term convection to advection.
V
2.2 Conservative Form of the Flow Equations
2.2.1 Mass (Continuity)
Physical principle ( mass conservation ): mass is neither created nor destroyed.
Consider an arbitrary control volume (aka “ cell ”). If the cell has
volume V and a typical cell face has area A and component of velocity
un along the (outward) normal:
mass of fluid: V
mass flux through one face: unA = u • A
Then
d
d mass + netoutwardmass flux = t
or
d
d
V
t
u A (4)
An equivalent differential equation for mass conservation can be
derived by considering a small Cartesian control volume with sides
x , y , z as shown.
d
d( )
- − + − + − = 144 444444444244 4444444443 (^12 3) netoutwardmass flux
e w n s t b
rate of change of mass
uA uA vA vA wA wA t
V
where density and velocity are averages over cell volume or cell face.
Noting that volume V = x y z and areas Aw = Ae = y z etc,
[( ) ( ) ] [( ) ( )] [( ) ( ) ] 0
d
d( )
- u − u y z + v − v z x + w − w x y = t
x y z e w n s t b
Dividing by the volume, ∆ x ∆ y ∆ z :
d
d
z
w w
y
v v
x
u u
t
e w n s t b
Proceeding to the limit x , y , z → 0:
z
w
y
v
x
u
t
This analysis is analogous to the finite-volume procedure, except that in the latter the control
volume does not shrink to zero; i.e. it is a finite- volume not infinitesimal- volume approach.
∆ x
∆y
∆z
s e
w n
b
t
V
A
u
un
2.2.2 Momentum
Physical principle ( Newton’s Second Law ): rate of change of momentum = force
The total rate of change of momentum for fluid passing through a control volume consists of:
- time rate of change of total momentum inside the control volume; plus
- net momentum flux (difference between rate at which momentum leaves and enters).
For a cell with volume V and a typical face with area A :
momentum in cell = mass × u =( V ) u
momentum flux = mass flux × u =( u • A ) u
momentum netoutwardmomentumflux force t
d
d (9)
or
( × u )+∑ ( × u )= F d
d
faces
mass mass flux t
Remember that momentum, velocity and force are vectors. There are strictly 3 component
equations.
Fluid Forces
There are two main types:
- surface forces (proportional to area; act on control-volume faces)
- body forces (proportional to volume)
(i) Surface forces are usually expressed in terms of stress (= force per unit area):
area
force stress = or force = stress × area
The main surface forces are:
- pressure p : always acts normal to a surface;
- viscous stresses : frictional forces arising from relative motion.
For a simple shear flow there is only one non-zero stress
component:
y
u
∂
but, in general, is a symmetric tensor (the components of
stress imparted by external fluid on individual faces of a
volume of fluid are shown right) and has a more complex
expression for its components. In incompressible flow:
i
j
j
i ij x
u
x
u
∂
τ 11
τ 21
τ 12
τ 22
τ 11
τ 21
τ 22
12
x
y
τ
V
A
u
un
y
U
(ii) Body forces
The main body forces are:
gravity : the force per unit volume is
g = ( 0 , 0 ,− g )
(For constant-density fluids, pressure and weight can be combined as a piezometric
pressure p
= p + gz ; with gravity incorporated into a modified pressure it does not then appear explicitly in the flow equations – see the Examples.)
- centrifugal and Coriolis forces (apparent forces in a rotating reference frame)
centrifugal force:
r R
2 − ∧( ∧ ) =
Coriolis force:
− 2 ∧ u
(Because the centrifugal force can be written as the gradient of
2 2 2
(^1) R it can also be
absorbed into a modified pressure and hence removed from the momentum equation;
see the Examples).
Differential Equation For Momentum
Once again, a differential equation can be derived by considering a
fixed Cartesian control volume with sides x , y and z.
For the x -component of momentum:
p A pA viscous and other fo rces
Vu uA u uA u vA u vA u wA u wA u t
pressureforceinx direction
w w e e
netoutwardmomentumflux
e e w w n n s s t t b b
rate of change of momentum
= − +
d
d
Substituting cell dimensions:
p p y z viscous and other fo rces
x y zu u u u u y z v u v u z x w u w u x y t
w e
e e w w n n s s t t b b
( ) [( ) ( ) ] [( ) ( ) ] [( ) ( ) ]
d
d
Dividing by volume x y z (and changing the order of pe and pw ):
ρg
z
∆ x
∆y
∆z
s e
w n
b
t
ΩΩΩΩ
R
r
axis
ρΩ R
2
u
ΩΩΩΩ
In inertial frame In rotating frame
2.2.3 General Scalar
A similar equation may be derived for any physical quantity that is advected or diffused by a
fluid flow. For each such quantity an equation is solved for the concentration (i.e. amount per
unit mass) φ: for example, the concentration of salt, sediment or a chemical constituent.
Diffusion occurs when concentration varies with position. It typically involves transport from
regions of high concentration to regions of low concentration, at a rate proportional to area
and concentration gradient. For many scalars it may be quantified by Fick ’ s diffusion law :
A
n
rate of diffusion diffusivity gradient area
∂ φ =−
=− × ×
This is often referred to as gradient diffusion. A common example is heat conduction.
For an arbitrary control volume:
amount in cell : V φ (mass × concentration)
advective flux : ( u • A )φ (mass flux × concentration)
diffusive flux : A ∂ n
∂ φ − (–diffusivity × gradient × area)
source S = sV (source density × volume)
Balancing the rate of change, the net flux through the boundary and rate of production yields
the scalar-transport or ( advection-diffusion ) equation :
rate of change + netoutward flux = source
A S
n
mass massflux t (^) faces
∂ φ ( × φ)+∑ ( ×φ − ) d
d (16)
or, in differential form,
s z
w y z
v x y
u t x
∂ φ φ − ∂
∂ φ φ − ∂
∂ φ φ − ∂
∂ φ ( ) ( ) ( )
(*** Advanced ***)
This may be expressed more mathematically as:
^ =
+ φ−Γ∇φ)• ⌡
φ V ∂ V V
V s V t
d ( d d d
d u A (18)
For a fixed control volume, taking the time derivative under the integral sign and using
Gauss’s divergence theorem as before gives a corresponding differential equation
s t
+∇• φ− ∇φ = ∂
∂ φ ( )
u (19)
V
A
u
un
2.2.4 Momentum Components as Transported Scalars
In the momentum equation, if the viscous force A = ( ∂ u /∂ n ) A is transferred to the LHS it
looks like a diffusive flux: e.g. for the x -component:
A other forces n
u mass u massflux u t (^) faces
( × )+∑ ( × − ) d
d
Compare this with the generic scalar-transport equation:
A S
n
mass massflux t (^) faces
∂ φ ( × φ)+∑ ( ×φ − ) d
d
Each component of momentum satisfies its own scalar-transport equation with
concentration ↔↔↔↔ velocity ( u , v or w )
diffusivity (^) ↔↔↔↔ viscosity
source ↔↔↔↔ other forces
Consequently, only one generic scalar-transport equation need be considered.
In Section 5 we shall see, however, that the momentum equations are slightly different from
passive scalars (those not affecting the flow), because they are:
- nonlinear (the mass flux involves the velocity component being solved for);
- coupled (the mass flux involves the other velocity components);
- pressure-linked.
2.2.5 Non-Gradient Diffusion
The analysis above assumes that all non-advective flux is simple gradient diffusion:
A
∂ n
∂ φ −
Actually, the real situation is a little more complex. For example, in the u -momentum
equation the full expression for the 1-component of viscous stress through the 2-face is
x
v
y
u 12
The u / y part is gradient diffusion of u , but the v / x term is not. In general, non-advective
fluxes F ′^ that can’t be represented by gradient diffusion are discretised conservatively (i.e.
worked out for particular cell faces) but are transferred to the RHS as a source term:
A F S
n
mass massflux t (^) faces
∂ φ ( × φ)+∑ [ ×φ − ] d
d
2.2.6 Moving Control Volumes
Control-volume equations are also applicable to moving control volumes, provided the
normal velocity component in the mass flux is that relative to the mesh ; i.e.
un = ( u − u mesh )• n
The finite-volume method can thus be used for calculating flows with moving boundaries
2 .
2 See, for example: Apsley, D.D. and Hu, W., 2003, CFD Simulation of two- and three-dimensional free-surface
flow, International Journal for Numerical Methods in fluids, 42, 465-491.
The material derivative is obtained for the particular path following the flow (d x /d t = u , etc.):
z
w y
v x
u t t ∂
∂ φ
∂
∂ φ
∂
∂ φ
∂
∂ φ ≡
φ
D
D
Using this definition, it is possible to write a non-conservative but more compact form of the
governing equations. For a general scalar φ the sum of time-dependent and advective terms in
its transport equation is
z
w
y
v
x
u
t ∂
∂ φ
∂
∂ φ
∂
∂ φ
∂
∂ ( φ) ( ) ( ) ( )
∂φ φ+ ∂
∂φ φ+ ∂
∂φ φ+ ∂
∂φ φ+ ∂
z
w z
w
y
v y
v
x
u x
u
t t
(by the product rule)
bycontinuity tbydefinition
z
w y
v x
u z t
w
y
v
x
u
t
0 D/ D
= = φ
∂φ
∂
∂φ
∂
∂φ
∂
∂φ φ +
(by collecting similar terms)
D t
Dφ = (21)
Using the material derivative, the time-dependent and advection terms in a scalar-transport
equation can be combined as the much more compact (but non-conservative) form Dφ/D t.
The material derivative of velocity (D u /D t ) is the acceleration. The momentum equation can
be written
u other forc es x
p
t
u
massaccelerati on
×
2
D
D
This form is simpler to write and is used both for convenience and to derive theoretical
results in special cases (see the Examples). However, in the finite-volume method it is the
conservative form which is actually discretised directly.
(*** Advanced ***)
The material derivative can be defined in suffix or vector notation as
i x
u t t ∂
∂ φ
∂
∂ φ
φ
D
D
or + •∇φ ∂
∂ φ
φ u D t t
D
Simplify the derivation of (21) above using the summation convention or vector notation.
2.4 Non-Dimensionalisation
Although it is possible to work entirely in dimensional quantities, there are good theoretical
reasons for working in non-dimensional variables. These include the following.
- All dynamically-similar problems (same Re, Fr etc.) can be solved with a single
computation.
- The number of relevant parameters (and hence the number of graphs) is reduced.
- It indicates the relative size of different terms in the governing equations and, in
particular, which might conveniently be neglected.
- Computational variables are the same order of magnitude, yielding better numerical
accuracy.
2.4.1 Non-Dimensionalising the Governing Equations
For incompressible flow the governing equations are:
continuity : = 0 ∂
z
w
y
v
x
u (23)
momentum : u x
p
t
u (^) 2
D
D
= − (and similar equations in y , z directions) (24)
Adopting reference scales U 0 , L 0 and 0 for velocity, length and density, respectively, and
derived scales L 0 / U 0 for time and
2 0 U^ 0 for pressure, each fluid quantity can be written as a
product of a dimensional scale and a non-dimensional variable (indicated by an asterisk *):
2 0 0 0 0 0
0 0 t U p p U p etc U
L
x = L x t = u = u = − ref =
(Note: In incompressible flow it is differences in pressure that are important, not absolute
values. Since these differences are usually much smaller than the absolute pressure it is
numerically more accurate to work in terms of the departure from a reference pressure pref ).
Substituting these into mass and momentum equations, (23) and (24), yields, after some
algebra:
= ∂
z
w
y
v
x
u (25)
2
Re
D
D
u x
p
t
u
= − where Re
0 U 0 L 0
From this, it is seen that the key dimensionless group is the Reynolds number Re. If Re is
large then viscous forces would be expected to be negligible in much of the flow.
Having derived the non-dimensional equations it is usual to drop the asterisks and simply
declare that you are “working in non-dimensional variables”.
Summary
- Fluid dynamics is governed by conservation equations for mass, momentum, energy
(and, for a non-homogeneous fluid, the amount of individual constituents).
- The governing equations can be expressed in equivalent integral (control-volume) or
differential forms.
- The finite-volume method is a direct discretisation of the control-volume equations.
- Differential forms of the flow equations may be conservative (i.e. can be integrated
directly to something of the form “ fluxout – fluxin = source ”) or non-conservative.
- A particular control-volume equation takes the form:
rate of change + net outward flux = source
- There are really just two canonical equations to discretise and solve:
mass conservation ( continuity ):
d
d
mass mass flux t
scalar-transport (or advection-diffusion ) equation :
rateof change advection diffusion source
A S
n
mass mass flux t (^) faces
∂ φ ( × φ) + ∑ ( ×φ − ) d
d
- Each Cartesian velocity component ( u, v, w ) satisfies its own scalar-transport
equation. However, these equations differ from those for a passive scalar because they
are non-linear and strongly coupled through the advective fluxes and pressure forces.
- Non-dimensionalising the governing equations, allows dynamically-similar flows
(those with the same values of Reynolds number, etc.) to be solved with a single
calculation, reduces the overall number of parameters, indicates when certain terms in
the governing equations are significant or negligible and ensures that the main computational variables are of similar magnitude.
Examples
Q1.
In 2-d flow the continuity and x -momentum equations can be written in conservative form
(and with compressibility neglected in the viscous forces) as
v y
u t x
u x
p vu y
uu x
u t
2 ( ) ( ) ( ) + ∇ ∂
respectively.
(a) By expanding derivatives of products show that these can be written in the equivalent
non-conservative forms:
D
D
y
v
x
u
t
u x
p
t
u (^) 2
D
D
where the material derivative is given (in 2 dimensions) by y
v x
u t t ∂
D
D
(b) Define what is meant by the statement that a flow is incompressible. To what does the
continuity equation reduce in incompressible flow?
(c) Write down conservative forms of the 3-d equations for mass and x -momentum.
(d) Write down the z -momentum equation, including gravitational forces.
(e) Show that, for constant-density flows, pressure and gravity can be combined in the
momentum equations as the piezometric pressure p + g z.
( *** Advanced *** )
(f) Write the conservative mass and momentum equations in vector notation.
(g) Write the conservative mass and momentum equations in suffix notation using the
summation convention.
(h) In a rotating reference frame there are additional apparent forces (per unit volume):
centrifugal force: − ∧( ∧ r ) or R
2
Coriolis force: − 2 ∧ u
where is the angular velocity vector (with magnitude and
direction along the axis of rotation), u is the fluid velocity in the
rotating reference frame, r is the position vector (relative to a point
on the axis of rotation) and R is a vector perpendicularly outward
from the axis of rotation to the point. By writing the centrifugal force as the gradient of some quantity show that it can be
subsumed into a modified pressure. Also, find the components of
the Coriolis force if rotation is about the z axis.
R
r
axis
ρΩ R
2
Q5. (Computational Hydraulics Exam, May 2008; parts (g) and (h) depend on later sections
of this course).
The momentum equation for a viscous fluid in a rotating reference frame is
u u
u = −∇ + ∇ − 2 ∧ D
D 2
p t
where is density, u = ( u , v , w ) is velocity, p is pressure, is dynamic viscosity and is the
angular-velocity vector of the reference frame. The symbol ∧ denotes a vector product.
(a) If = ( 0 , 0 , )write the x and y components of the Coriolis force ( − 2 ∧ u ).
(b) Hence write the x - and y -components of equation (*).
(c) Show how Equation (*) can be written in non-dimensional form in terms of a
Reynolds number Re and Rossby number Ro (both of which should be defined).
(d) Define the terms conservative and non-conservative when applied to the differential
equations describing fluid flow.
(e) Define (mathematically) the material derivative operator D/D t. Then, noting that the
continuity equation can be written
z
w
y
v
x
u
t
show that the x -momentum equation can be written in an equivalent conservative
form.
(f) If the x -momentum equation were to be regarded as a special case of the general
scalar-transport (or advection-diffusion) equation, identify the quantities representing: (i) concentration;
(ii) diffusivity;
(iii) source.
(g) Explain why the three equations for the components of momentum cannot be treated
as independent scalar equations.
(h) Explain (briefly) how pressure can be derived in a CFD simulation of:
(i) high-speed compressible gas flow;
(ii) incompressible flow.
Q6. (MSc Exam, May 2011 – part)
(a) In a rotating reference frame (with angular velocity vector ) the non-viscous forces
on a fluid are, per unit volume,
(I) (II) (III) (IV)
2 − ∇ p + g + R − ∧ u
where p is pressure, g = (0,0,– g ) is the gravity vector and R is the vector from the
closest point on the axis of rotation to a point. Show that, in a constant-density fluid,
forces (II) and (III) can be absorbed into a modified pressure force.
(b) Consider a closed cylindrical can of radius 5 cm and depth 15 cm. The can is
completely filled with fluid of density 1100 kg m
- and is rotating steadily about its
axis (which is vertical) at 600 rpm. Where do the maximum and minimum pressures
in the can occur, and what is the difference in pressure between them?
Q7. (Computational Hydraulics Exam, May 2011 – part)
The figure below depicts a 2-d cell in a finite-volume CFD calculation. Vertices are given in
the figure, and velocity in the adjacent table. At this instant = 1.0 everywhere.
(a) Calculate the volume flux out of each face. (Assume unit depth.)
(b) Show that the flow is not incompressible and find the time derivative of density.
e
s
w
n
x
y
Face Velocity ( u , v )
u v
e 4 10
n 1 8
w 2 2
s 1 4
