me 231 thermofluid mechanics i navier-stokes equations

ME 231Thermofluid

Mechanics I

Navier-Stokes Equations

Computational Fluid Dynamics (AE/ME 339) K. M. Isaac

MAEEM Dept., UMR

Equations of Fluid Dynamics, Physical Meaning of the terms.

Equations are based on the following physical principles:

• Mass is conserved

• Newton’s Second Law:

• The First Law of thermodynamics: e = q - w, for a system.

F = m a


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Control Volume Analysis

The governing equations can be obtained in the integral form by choosing a control volume (CV) in the flow field and applying the principles of the conservation of mass, momentum and energy to the CV.


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MAEEM Dept., UMR

Consider a differential volume element dV in the flow field. dV is small enough to be considered infinitesimal but large enough to contain a large number of molecules for continuum approach to be valid.

dV may be:

• fixed in space with fluid flowing in and out of its surface or,

• moving so as to contain the same fluid particles all the time. In this case the boundaries may distort and the volume may change.


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Substantial derivative

(time rate of change following a moving fluid element)

1

2

x

y

z

Fluid element attime t = t1

Fluid element attime t = t2

V1

V2

i

j

k


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The velocity vector can be written in terms of its Cartesian components as:

where

u = u(t, x, y, z)

v = v(t, x, y, z)

w = w(t, x, y, z)

ˆˆ ˆ( , , , ) ( , , , ) ( , , , )V iu t x y z jv t x y z kw t x y z


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@ time t1:

@ time t2:

Using Taylor series

1 1 1 1 1( , , , )t x y z

2 2 2 2 2( , , , )t x y z

2 1 2 1 2 1 2 1 2 11 1 11

( ) ( ) ( ) ( )t t x x y y z zt x y z

( )higher order terms


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The time derivative can be written as shown on the RHS in the following equation. This way of writing helps explain the meaning of total derivative.

2 1 2 1 2 1 2 1

1 1 12 1 2 1 2 1 2 11

...........(2.1)x x y y z z

t t t x t t y t t z t t


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We can also write

2 1

2 1

2 1

limt t

D

t t Dt

2 1

2 1

2 1

limt t

x xu

t t

2 1

2 1

2 1

limt t

y yv

t t

2 1

2 1

2 1

limt t

z zw

t t


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where the operator can now be seen to be defined in the following manner.

...........(2.2)D

u v wDt t x y z

D

Dt

.........(2.3)D

u v wDt t x y z


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The operator in vector calculus is defined as

which can be used to write the total derivative as

ˆˆ ˆ ...........(2.4)i j kx y z

...........(2.5)D

VDt t


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Example: derivative of temperature, T

( ) ........(2.6)

convectivelocal derivativederivative

DT T T T T TV T u v w

Dt t t x y z


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A simpler way of writing the total derivative is as follows:

............(2.7)d dt dx dy dzt x y z

............(2.8)d dx dy dz

dt t x dt y dt z dt

..........(2.9)d

u v wdt t x y z


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The above equation shows that and have the same meaning,

and the latter form is used simply to emphasize the physical meaning that it consists of the local derivative and the convective derivatives. Divergence of Velocity (What does it mean?) Consider a control volume moving with the fluid. Its mass is fixed with respect to time.Its volume and surface change with time as it moves from one location to another.

d

dt

D

Dt


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Insert Figure 2.4


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The volume swept by the elemental area dS during time interval t can be written as

Note that, depending on the orientation of the surface element, v could be positive or negative. Dividing by t and letting 0 gives the following expression.

ˆ .........(2.10)V V t n dS V t dS

V

0

1..........(2.11)lim

t S S

DVV t dS V dS

Dt t


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The LHS term is written as a total time derivative because the fluid element is moving with the flow and it would undergo both the local acceleration and the convective acceleration. The divergence theorem from vector calculus can now be used to transform the surface integral into a volume integral.

. ........(2.12)V

DVV dV

Dt


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If we now shrink the moving control volume to an infinitesimal volume, , , the above equation becomes

When the volume integral can be replaced by on the RHS to get the following.

The divergence of is the rate of change of volume per unit volume.

δV

( )...............(2.13)

D VV dV

Dt

δV 0

VδV

V

1 ( ).........(2.14)

D VV

V Dt

δV


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MAEEM Dept., UMR

Continuity Equation

Consider the CV fixed in space. Unlike the earlier case the shape and size of the CV are the same at all times. The conservation of mass can be stated as: Net rate of outflow of mass from CV through surface S = time rate of decrease of mass inside the CV


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The net outflow of mass from the CV can be written as

Note that by convention is always pointing outward. Therefore can be (+) or (-) depending on the directions of the velocity and the surface element.

..........(2.16)nV dS V dS

V dSdS


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Total mass inside CV

Time rate of increase of mass inside CV (correct this equation)

Conservation of mass can now be used to write the following equation

See text for other ways of obtaining the same equation.

..........(2.20)V

Mass dV

................(2.18)V

dVt

0.................(2.19)V S

dV V dSt


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0........(2.21)V

DdV

Dt

Integral form of the conservation of mass equation thus becomes


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MAEEM Dept., UMR

An infinitesimally small element fixed in space


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Net mass flow =

( ) uuu dx dydz u dydz dxdydz

x x

v vv dy dxdz v dxdz dxdydz

y y

w ww dz dxdy w dxdy dxdydz

z z

..........(2.22)

u v wdxdydz

x y z

Net outflow in x-direction

Net outflow in y-direction

Net outflow in z-direction


MAEEM Dept., UMRvolume of the element = dx dy dz mass of the element = (dx dy dz)

Time rate of mass increase = ...........(2.23)dxdydzt

Net rate of outflow from CV = time rate of decrease of mass within CV

u v wdxdydz dxdydz

x y z t

0.......(2.24)

u v w

t x y z

or


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0..........(2.25)Vt

Which becomes

The above is the continuity equation valid for unsteady flow Note that for steady flow and unsteady incompressible flow the first term is zero.

Figure 2.6 (next slide) shows conservation and non-conservation forms of the continuity equation. Note an error in Figure 2.6: Dp/Dt should be replace with D/Dt.


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MAEEM Dept., UMRMomentum equation


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Consider the moving fluid element model shown in Figure 2.2bBasis is Newton’s 2nd Law which says F = m aNote that this is a vector equation. It can be written in terms of the three cartesian scalar components, thefirst of which becomes

Fx = m ax

Since we are considering a fluid element moving with the fluid, its mass, m,is fixed. The momentum equation will be obtained by writing expressions for theexternally applied force, Fx, on the fluid element and the acceleration, ax,of the fluid element. The externally applied forces can be divided into two types:


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1. Body forces: Distributed throughout the control volume. Therefore,this is proportional to the volume. Examples: gravitational forces, magnetic forces, electrostatic forces.

2. Surface forces: Distributed at the control volume surface. Proportionalto the surface area. Examples: forces due to surface and normal stresses.These can be calculated from stress-strain rate relations.

Body force on the fluid element = fx dx dy dzwhere fx is the body force per unit mass in the x-direction

The shear and normal stresses arise from the deformation of the fluidelement as it flows along. The shape as well as the volume of the fluidelement could change and the associated normal and tangential stressesgive rise to the surface stresses.


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MAEEM Dept., UMRThe relation between stress and rate of strain in a fluid is known from thetype of fluid we are dealing with. Most of our discussion will relate to Newtonian fluids for which

Stress is proportional to the rate of strain

For non-Newtonian fluids more complex relationships should be used.

Notation: stress ij indicates stress acting on a plane perpendicular to the i-direction (x-axis) and the stress acts in the direction, j, (y-axis).

The stresses on the various faces of the fluid element can written asshown in Figure 2.8. Note the use of Taylor series to write thestress components.


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The normal stresses also has the pressure term.

Net surface force acting in x direction =

pp p dx dydz

x

yxxxxx xx yx yxdx dydz dy dxdz

x y

.............(2.46)zxzx zxdz dxdy

z


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...........(2.47)yxxx zxx x

pF dxdydz f dxdydz

x x y z

........(2.48)m dxdydz

...........(2.49)x

Dua

Dt


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.........(2.50 )yxxx zxx

Du pf a

Dt x x y z

.........(2.50 )xy yy zyy

Dv pf b

Dt y x y z

.........(2.50 )yzxz zzz

Dw pf c

Dt z x y z


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....................(2.51)Du u

V uDt t

u uu

t t t

.............(2.52)

uuu

t t t

........(2.53)V u uV u V

uDuu u V uV

Dt t t

.............(2.54)u

u V uVt t


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The term in the brackets is zero (continuity equation)The above equation simplifies to

...........(2.55)uDu

uVDt t

Substitute Eq. (2.55) into Eq. (2.50a) shows how the following equationscan be obtained.

..........(2.56 )yxxx zxx

u puV f a

t x x y z

..........(2.56 )xy yy zyy

v pvV f b

t y x y z

..........(2.56 )yzxz zzz

w pwV f c

t z x y z


MAEEM Dept., UMRThe above are the Navier-Stokes equations in “conservation form.”


MAEEM Dept., UMRFor Newtonian fluids the stresses can be expressed as follows

2 ...........(2.57 )xx

uV a

x

2 ...........(2.57 )yy

vV b

y

2 ...........(2.57 )zz

wV c

z

...........(2.57 )xy yx

v ud

x y

...........(2.57 )xz zx

u we

z x

...........(2.57 )yz zy

w vf

y z


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In the above is the coefficient of dynamic viscosity and is the second viscosity coefficient.Stokes hypothesis given below can be used to relate the above two coefficients

= - 2/3 The above can be used to get the Navier-Stokes equations in the followingfamiliar form


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2

2uu uv uw p u

Vt x y z x x x

.........(2.58 )x

v u u wf a

y x y z z x

2

2vv uv vw p v

Vt y x z y y y

.........(2.58 )y

v u v wf b

x x y z z y


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2

2ww uw vw p w

Vt z x y z z z

.........(2.58 )z

w v u wf c

y y z x z x

The energy equation can also be derived in a similar manner.


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The above equations can be simplified for inviscid flows by droppingthe terms involving viscosity.

Summary

• Apply conservation equations to a control volume (CV)• As the CV shrinks to infinitesimal volume, the resulting partial

differential equations are the Navier-Stokes equations• Taylor series can be used to write the variables• The total derivative consists of local time derivative and convective

derivative terms• In incompressible flow, divergence of velocity is a statement of

the conservation of volume• Need surface and body forces to write the momentum equation• Surface forces are: pressure forces, forces due to normal stresses

and forces due to shear stresses• Body forces are due to weight, magnetism and electrostatics • Momentum equation is a vector equation. Can be written in terms

of its components.

Summary

• Stresses are indicated by a plane and a direction, respectively, by two subscripts• For Newtonian fluids, stresses are proportional to the rates of strain• Stokes hypothesis is used to relate the first and second coefficients of viscosity• The resulting equations are the Navier-Stokes equations• In order to solve the equations, they must be simplified for the problem

you are considering (e. g., boundary layer, jet, airfoil flow, nozzle flow)

me 231 thermofluid mechanics i navier-stokes equations

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