2. Fluid-Flow Equations, Schemes and Mind Maps of Fluid Mechanics

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2. Fluid-Flow Equations

Governing Equations

• Conservation equations for:

‒ mass

‒ momentum

‒ energy

‒ (other constituents)

• Alternative forms:

‒ integral (control-volume) equations

‒ differential equations

Mass Conservation (Continuity)

Mass flux through a face:

Mass in a cell:

Mass conservation : mass is neither created nor destroyed

V

A

u

u

n

d

d𝑡

(mass) = net inward mass flux

d

d𝑡

(mass) + net outward mass flux = 0

d

d𝑡

mass + ෍

faces

mass flux = 0

𝐶 = 𝜌 u • A

Mass Conservation - Differential Equation

 x

 y

 z

s

e

n

w

b

t

Divide by volume:

Shrink to a point:

d

d𝑡

mass + net outward mass flux = 0

d

d𝑡

𝑒

𝑤

𝑛

𝑠

𝑡

𝑏

d

d𝑡

𝜌Δ𝑥Δ𝑦Δ𝑧 + [(𝜌𝑢)

𝑒

𝑤

Δ𝑦Δ𝑧 + [(𝜌𝑣)

𝑛

𝑠

Δ𝑧Δ𝑥 + [(𝜌𝑤)

𝑡

𝑏

d𝜌

d𝑡

𝑒

𝑤

𝑛

𝑠

𝑡

𝑏

d𝜌

d𝑡

• ∇ • (𝜌 u ) = 0

Conservation statement:

Momentum Equation

Momentum Principle : force = rate of change of momentum

If steady : force = (momentum flux)

out

• (momentum flux)

in

If unsteady : force = d/d𝑡(momentum inside control volume)

+ (momentum flux)

out

• (momentum flux)

in

F

Momentum Equation

= (𝜌 u • A ) u

Momentum flux through a face

= (𝜌𝑉) u

Momentum of fluid in a cell

Rate of change of momentum = force

d

d𝑡

mass × u + ෍

faces

(mass flux × u ) = F

= mass × u

= mass flux × u

V

A

u

u

n

d

d𝑡

(momentum) + net outward momentum flux = force

Differential Equation

d

d𝑡

𝜌𝑉𝑢 + (𝜌𝑢𝐴)

𝑒

𝑢

𝑒

− (𝜌𝑢𝐴)

𝑤

𝑢

𝑤

• (𝜌𝑣𝐴)

𝑛

𝑢

𝑛

− (𝜌𝑣𝐴)

𝑠

𝑢

𝑠

• (𝜌𝑤𝐴)

𝑡

𝑢

𝑡

− (𝜌𝑤𝐴)

𝑏

𝑢

𝑏

= 𝑝

𝑤

𝐴

𝑤

− 𝑝

𝑒

𝐴

𝑒

• viscous and other forces

2

𝑢 + other forces

d(𝜌𝑢)

d𝑡

𝑒

𝑤

𝑛

𝑠

𝑡

𝑏

𝑒

𝑤

viscous and

other forces

d

d𝑡

(momentum) + net momentum flux = force

d

d𝑡

𝜌Δ𝑥Δ𝑦Δ𝑧 𝑢 + [(𝜌𝑢)

𝑒

𝑢

𝑒

− 𝜌𝑢)

𝑤

𝑢

𝑤

Δ𝑦Δ𝑧 + [(𝜌𝑣)

𝑛

𝑢

𝑛

− 𝜌𝑣)

𝑠

𝑢

𝑠

Δ𝑧Δ𝑥 + [(𝜌𝑤)

𝑡

𝑢

𝑡

− 𝜌𝑤)

𝑏

𝑢

𝑏

Δ𝑥Δ𝑦

= 𝑝

𝑤

− 𝑝

𝑒

Δ𝑦Δ𝑧 + viscous and other forces

 x

 y

 z

s

e

n

w

b

t

Divide by volume:

Shrink to a point:

d(𝜌𝑢)

d𝑡

viscous and

other forces

Conservation statement:

General Scalar

Source :

d

d𝑡

mass × 𝜙 + ෍

faces

(mass flux × 𝜙 − Γ

Amount in a cell:

(mass  concentration)

‒ diffusion :

Flux through a face:

‒ advection : (𝜌 u • A )𝜙

(mass flux  concentration)

(diffusivity  gradient  area)

Time derivative + net outward flux = source

𝜙 = concentration (amount per unit mass)

V

A

u

u

n

(source density  volume)

Differential Equations For Fluid Flow

Forms of the equations in primitive variables may be:

• Conservative

‒ can be integrated directly to give “net flux = source”

• Non-conservative

‒ can’t be integrated directly

Other forms of the equations include those for:

• Derived variables

‒ e.g. velocity potential

Example

d

d𝑥

(𝑦

) = 𝑔(𝑥)

2𝑦

d𝑦

d𝑥

= 𝑔(𝑥)

conservative

non-conservative

Same equation! ... but only the first can be integrated directly

Non-Conservative Flow Equations

D𝑢

D𝑡

mass × acceleration

2

forces

e.g. momentum equation:

conservative form

non-conservative form

Proof:

= 0 by continuity

D𝜙

D𝑡

=D𝜙/D𝑡 by definition

D𝜙

D𝑡

(mass conservation)

Example Q1 (Equation Manipulation)

In 2-d flow, the continuity and x - momentum equations can be written in conservative form as

(a) Show that these can be written in the equivalent non-conservative forms:

(b) Define carefully 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 the gravitational force.

(e) Show that, for constant-density flows, pressure and gravity can be combined in the momentum equations via the

piezometric pressure 𝑝 + 𝜌𝑔𝑧.

(f) In a rotating reference frame there are additional apparent forces (per unit volume):

centrifugal force:

Coriolis force:

where Ω is the angular velocity of the reference frame, u is the fluid velocity in that frame, r is the position vector

and R is its projection perpendicular to the axis of rotation. 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.

𝜕𝜌

𝜕𝑡

𝜕

𝜕𝑥

(𝜌𝑢) +

𝜕

𝜕𝑦

(𝜌𝑣) = 0

𝜕

𝜕𝑡

(𝜌𝑢) +

𝜕

𝜕𝑥

(𝜌𝑢𝑢) +

𝜕

𝜕𝑦

(𝜌𝑣𝑢) = −

𝜕𝑝

𝜕𝑥

• 𝜇∇

2

𝑢

D𝜌

D𝑡

• 𝜌(

𝜕𝑢

𝜕𝑥

𝜕𝑣

𝜕𝑦

) = 0

𝜌

D𝑢

D𝑡

= −

𝜕𝑝

𝜕𝑥

• 𝜇∇

2

𝑢

𝜌Ω

2

R

−2𝜌 Ω ∧ u



R

r

axis

  R

2

(b) Define carefully what is meant by the statement that a flow is incompressible. To what does the

continuity equation reduce in incompressible flow?

Incompressible : flow-induced changes to pressure (or temperature)

do not cause significant changes in density

D𝜌

D𝑡

D𝜌

D𝑡

(c) Write down conservative forms of the 3-d equations for mass and x - momentum.

2

2 - d continuity:

2 - d x - momentum:

2

2

2

𝜕

2

𝜕𝑦

2

2

3 - d x - momentum: ∇

2

2

2

𝜕

2

𝜕𝑦

2

𝜕

2

𝜕𝑧

2

2

3 - d z - momentum:

Pressure + gravity forces:

− ρ𝑔

∗

∗

∗

∗

(d) Write down the 𝑧-momentum equation, including the gravitational force.

(e) Show that, for constant-density flows, pressure and gravity can be combined in the momentum

equations via the piezometric pressure 𝑝 + ρ𝑔𝑧.

3 - d continuity: