ns equations

8/12/2019 NS Equations

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VII. Derivation of the Navier-Stokes Equations and Solutions

In this chapter, we will derive the equations governing 2-D, unsteady, compressibleviscous flows. These equations (and their -D form! are called the "avier-#to$es equations. Theywere developed by "avier in %&%, and more rigorously be #to$es in %&'. "ow, over %) yearslater, these equations still stand with no modifications, and form the basis of all simpler forms of

equations such as the potential flow equations that were derived in *hapter I.

In two dimensions, we have five flow properties that are un$nowns+ the two velocity

components u,v density, temperature T and pressure p. Therefore, we need equations lin$ing

them. ne of these equations is the equation of state, given by

p =p(,T )

t moderate temperatures that arise in subsonic and supersonic flows without chemicalreactions, this equation of state may be simplified to the following form+

p = RT

/ere 0 is a gas constant, given by R/M, where Ris the universal gas constant, and Misthe molecular weight of the gas (or the gas mi1ture!. or air, the gas constant is given by 03 2&%45oules6$g67 8.

The other four equations are+

a! *onservation of mass, $nown as continuity,

b! *onservation of u- momentum

c! *onservation of v- momentum, and

d! *onservation of energy.

*onservation of 9ass

:e consider a small control volume (*;! of height y, width 1, and of depth unity

perpendicular to the plane of the paper.


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1

y


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"ow, consider the limits of the above equation as 1 and y goes to Aero. rom calculus,

for any arbitrary function f(1,y!,

fx

=Limit x0f x + x,y( )f x,y( )

x

fy

=Limit y0 f x,y + y( )f x, y( )y

pplying the above limits, and bringing all the terms to the left hand side, we get

t

+ u( )

x+

v( )y

= 0

The above equation, in vector form is given by+

t + V( )=0where,

= ' Del' Operator =x

i +y

j

and,

V =ui +vj

The vector form is more useful than it would first appear. If we want to derive thecontinuity equation in another coordinate system such as the polar, cylindrical or sphericalcoordinate system, all we need to $now is (a! loo$ up the BDelB operator in that system, (b! loo$ up

the rules for the dot product of BDelB operator and a vector in that system, (c! perform the dotproduct.

*onservation of u- 9omentum Cquation

average> over a period of time is called the flowvelocity component u, and is measured by probes such as ?D;s and hot wires.

(ii! C1change of u- momentum between the molecules on the left and those on the right bycollisions. In this case, there no net gain in mass, but there is a gain (or loss! in momentum.These collision effects may be averaged over a sufficiently small period of time, and may beviewed as a pressure force e1erted by the fluid on the left on our *;. gain, only this averageeffect is felt or measured by pressure probes, and barometers. The individual collisions occur fartoo rapidly and far too frequently to be sensed by probes or measuring devices.


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(iii! FC1change of u- and v- momentum by random linear motion of molecules umping in andout of our control volume, across the face %. In this case, all the molecules that umped in also

ump out over a sufficiently small time period. Thus, this random motion does not add mass to ourcontrol volume (and was not considered in our >continuity> equation!. They however bring u- andv-momentum in or out (associated with their random motion!. The time averages of these rates at

which u- and v- momentum is brought into the *; across a face are called viscous forces. Theforces per unit area are called viscous stresses. The viscous stresses that bring in6out u-momentum are called normal viscous stresses, while those that bring in v- momentum (byentering the face at an angle! are calledtangential viscous stresses.

*onvection>

measured by Grobes

such as ?D;,

/ot wire

C1change of u- momentum

by random collisions

over a small period of time>*alled Gressure>

9easured by pressure

probes

0andom, bac$ and forth

linar motion of molecules

that bring in both the

u- and v- momentum

into the *;

p3

"ormal

#hear

3

These stresses (normal, and tangential or shear! are given the symbol . They are

identified by two subscripts.

(i! The first subscript indicates the plane on which they act. or e1ample, if a plane is normal tothe 1- a1is, the first subscript will be 1.

(ii! The second subscript identifies the direction of the force associated with the force. ore1ample, if a shear force is pointing in the y- direction, the second subscript will be y.


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Thus, viscous stress is a tensor quantity, and requires three pieces of information (itsmagnitude, its direction and the plane on which it acts! to completely specify it. This separates atensor from a vector (magnitude and direction!, and a scalar (magnitude only!.

1

y

"ewtonian luids


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V

hThe proportionality factor turned out to be a constant at moderate temperatures, and was

called the coefficient of viscosity, . urthermore, for this particular case, the velocity profile is

linear, giving ;6h 3 Hu6Hy. Therefore, "ewton postulated+

= u

yluids that have a linear relationship between stress and strain rate are called Newtonian

fluids. This is a property of the fluid, not the flow. :ater and air are e1amples of "ewtonian fluids,while blood is a non-"ewtonian fluid.

#to$es /ypothesis+

#to$es e1tended "ewtonBs idea from simple %-D flows (where only one component ofvelocity is present! to multidimensional flows. here, the fluid element may e1perience a strain rateboth due to gradients such as Hu6Hy as well as Hv6H1. /e developed the following relations,collectively $nown as Stokes relations.

xx =2ux

+ ux

+vy

+wz

yy =2vy

+ ux

+vy

+wz

zz =2wz

+ ux

+vy

+wz

xy = yx = u

y+

v

x

xz = zx = uz

+wx

zy = yz = wy

+ vz

These e1pressions hold for -D flows. or 2-D flows, somewhat simpler e1pressions areobtained if we set w, the A- component of velocity, to Aero, and if we set all derivatives withrespect to A to be Aero.

The quantity is called the molecular viscosity, and is a wea$ function of temperature.or air viscosity increases with temperature, because viscous effects are associated with random

molecular motion. The coefficient was chosen by #to$es so that the sum of the normal stresses

11

, yy

and AA

are Aero. Then

= 2

3


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The above equation, and the requirement that the three normal stresses add up to Aeroare called #to$es hypothesis.

0eturning


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#umming up these contributions, dividing through 1y and ta$ing the limits as 1 and y

go to Aero, we get+

u( )

t+

u2 + p( )x

+ uv( )

y=

xx

x+

xy

y

Derivation of v- 9omentum Cquation+

The v- momentum equation may be derived using a logic identical to that used above,and is left as an e1ercise to the student. The final form is+

v( )t

+ uv( )

x+

v2 + p( )y

=xyx

+yyy

Derivation of the Cnergy Cquation+

The energy equation is a generaliAed form of the first law of Thermodynamics (that youstudied in 9C22 and C ))'!. The only difference here is that we are studying an opensystem (i.e. control volume! that can gain and lose mass. The classical form studied in courseson thermodynamics is applicable only for closed systems - i.e. fi1ed collection of particles.

The first law of Thermodynamics states that

>(! The rate at which the total (i.e. internal @ $inetic! energy increases within acontrol volume is equal to

(

(*! the rate at which wor$ is done on the control volume boundary by surface forces@

(D! the rate at which wor$ is done on the *; by body forces @

(C! the rate at which heat is added to the control volume at the surfaces by heatconduction @

(! the rate at which heat is released is added within the *; due to chemicalreactions.>

In our derivation, we will neglect the term (D! which corresponds to the wor$ done by the

body forces (such as gravity, electrical and electromagnetic forces! and (! which corresponds tochemical reactions. In some applications, e.g. modeling of weather, term D is important, while inothers (e.g. modeling combustors! term is important.

:e define the specific total energy (i.e. total energy per unit mass! C as

E =e +u2 +v2

2=CvT +

u2 +v2

2


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In the above equation BeB is specific internal energy. The word BspecificB means Bper unitmassB.

Term +Then, term may be e1pressed as follows+

Rate at which total energy increases within the CV = E( )

txy

Term


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pu xxu xyv( )y1 pu xxu xyv( )y

2+ pv yxu yyv( )x

3 pv yxu yyv( )y

4

= xypu xxu xyv( )

1 pu xxu xyv( )

2

x

+ xypv yxu yyv( )

3 pv yxu yyv( )

1

y

xy x

pu xxu xyv( ) xy y pv yxu yyv( )Term D+

The wor$ done by body forces is neglected here.

Term C+

The heat conduction effects are associated with the random motion of gas moleculesacross the control volume. s they move in and out, they bring energy into and out of the control

volume. :hen integrated over a small but finite period of time, a net e1change of heat energyoccurs at the boundary, without any e1change in mass. #ince this process is a random, chaoticprocess, it must somehow be empirically modeled. :e adapt ourierBs law used to modelconduction of heat through solids, which states

>The rate at which heat flows across a surface of unit area is proportional to the negative of thetemperature gradient normal to this surface>.

The constant of proportionality is called conductivity , a property of the solid (or fluid, in

the present conte1t!. "ote that the heat flu1 is proportional to the negative of the gradient,because heat flows from hot to cold.

:e can sum up the heat conduction effects at all the four boundaries. The result is,

Tx

y1

+ Tx

y2

Ty

x3

+ Ty

x4

= xyT

x 2 T

x 1x

+ xy

Ty

4

Ty

3

y

xy

T

x

x + xy

T

y

y

#umming up all the terms through C, and dropping the common factor 1y that

appears everywhere, we get the energy equation.


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E( )t

+ uE( )

x+

vE( )y

+ up( )

x+

vp( )y

=

uxx +vxy( )x

+ uyx +vyy( )

y+

T /x( )x

+ T /y( )

y

Grandtl "umber+ The coefficient of conductivity $ is a property of the fluid, li$e the molecularviscosity. It is also a wea$ function of temperature and so is the molecular viscosity.


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v( )t

+ uv( )

x+

v2 + p( )y

=

xy

x+

yy

y

Cnergy Cquation+

e +

u2 +v2

2

t

+ uh0( )x

+ vh0( )y

=

kTx

x+

kTy

y

+uxx +vxy( )

x+

uxy +vyy( )y

/ere.

3 density u,v 3 *artesian *omponents of velocity along 1,y a1es

p 3 Gressure T 3 Temperature.

lso,

e 3 #pecific Internal Cnergy 3 Internal energy per unit mass of the fluid 3 *vT , where *

vis the

specific heat at constant volume.

h)

3 #pecific Total Cnthalpy 3 Total enthalpy per unit mass of the fluid 3 *pT@(u

2@v

2!62 , where

*p is the specific heat at constant pressure.

inally, the viscous stresses are related to the velocity field by #to$es relations

xx

= 2ux

2

3

ux

+vy

yy

= 2vy

2

3

ux

+vy

xy

= yx

= uy

+ vx

The molecular viscosity and conductivity $ are properties of the fluid and are functions

of temperature. These two quantities are related by the Grandtl number

Pr =Cp

k

or air, the Grandtl number is around ).42 at room temperatures.


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#implified orm for #teady, 2-D Incompressible lows+In steady, incompressible flows, we can drop the time derivatives because the flow is

steady. The density may be also assumed constant. Then, the first three equations in the full"avier-#to$es equations set become+

ux

+ vy

= 0

uux

+ vuy

+1

px

= 2ux2

+2uy2

uvx

+ vvy

+ 1

py

= 2v

x2+

2v

y2

whereis called the $inematic viscosity3/

Nondimensionalization of the Viscous Flow Equations

*onsider the 2-D viscous flow past an airfoil shown below+

t first glance it appears that such a flow will depend on a large number of parameterssuch as (a! airfoil shape, including its chord c ,(b! its angle of attac$, (c! the freestream

temperature T

(d! the freestream density

, (e! the freestream velocity ;

, (f! the freestream

viscosity

, (g! the freestream conductivity $

, and so on.


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:e wish to $now if these large number of physical and geometric parameters can begrouped into a handful of parameters that can be systematically varied to study their effect on theflow. There are two common ways of identifying these parameters+

(i! Dimensional analysis+ /ere we attempt to combine the parameters listed about toarrive at a nondimensional form. or e1ample, after some trial and error, we can show that the

quantity ;c6 is a nondimensional quantity. Intuition and e1perience are needed to realiAe

that this nondimensional parameter, called 0eynolds number, is also a useful physical parameter.

Dimensional analysis will also produce combinations such asp c / . This parameter,

while nondimensional, is not a significant parameter in incompressible flows, as e1perienceshows.

(ii! "ondimensionaliAation of Joverning Cquations+ This approach provides a formalmanner by which nondimensional parameters of physical significance may be uncovered. Thisapproach is usually used in combination with the dimensional analysis shown above. To illustratehow this approach wor$s, let us consider the 2- flow over an airfoil shown above.

?et us introduce nondimensional quantities identified with a prime (B! as follows+

=

; x =x

c;y' =

y

c; t'=

tV

c;p' =

p

V2 ;' =

:hen these nondimensional quantities are used to replace the corresponding physicalquantities in the continuity equation the following form results+

't'

+ ' u'( )

x'+

' v'( )y'

=0

This form loo$s identical to the dimensional form given in /andout =%, and no newnondimensional parameter emerges.

If we repeat this process with the u- momentum equation, and replace terms such as 1y

and p with terms such as

xy

= uy + vx( )= Vc

'

u'y'

+ v'x'

p = V2p'

we get, after some minor algebra+


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' u'( )t'

+ ' u' 2 +p'( )

x'+

' u' v'( )y'

= 1Re

xx

'

x'+

xy

'

y'

where,

Re = Reynolds Number =Vc

If we repeat this process with the v- momentum equation, again the 0eynolds numberemerges as the nondimensional parameter.

*onsider two different flows over the same airfoil, but of different chord. The presentnormaliAation shows that these two flows are governed by the same nondimensional form of thegoverning equations, and will result in the same nondimensional flow quantities such as uB, vB etc.

if the flows are geometrically similar (same airfoil shape, same !, and dynamically similar (same

0eynolds number!.

The parameter 0eynolds number may also be interpreted in another way. *onsider a

typical inviscid term such as p appearing on the left side of the u- momentum equation. This>inertial> force term is roughly of order V

2

. *onsider a viscous stress term such as 1y

. This

term is of order

Vc . Then, 0eynolds number is a measure of the ratio between the inertial

forces and viscous forces acting on a fluid element. If this number is large, then inertial forcesdominate over viscous forces and vice versa.

In practical aeronautical applications, the 0eynolds number is invariably large. ore1ample, the airfoil over a helicopter rotor blade operates at a 0eynolds number of to 9illion,while the airfoil in the 0oot section of a modern transport aircraft operates at a 0eynolds numberof ) to ) 9illion. In such flows, viscous forces are small and may be neglected over most of theflow, e1cept in very small regions called boundary layers over the airfoil.

"ondimensionaliAation of the Cnergy Cquation+ If we nondimensionaliAe the energy equation by

the same principles, we get, in addition to the 0eynolds number, the parameterCp/ k. This

parameter occurs due to the presence of the heat conduction terms and viscous stress wor$terms in the energy equation. This parameter is called the Grandtl number, and is a property ofthe fluid. or air, Grandtl number is around ).42.

In compressible flows, the quantity p/V2

is a nontrivial quanity and must beprescribed. Ksing equation of state, the definition of the speed of sound and the definition of

9ach number, we can relate this quantity to the 9ach number and the ratio of specific heats, .

p = RTa = RTM =V/ a

p

V2 =

1

M 2

Thus, in compressible flows, two flows are identical if they are geometrically similar and if0eynolds number, 9ach number, Grandtl number and ratio of specific heats all match.


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In incompressible flows, this quantity p/V2

is not a significant parameter. shydraulics engineers will attest, raising the static pressure at some point in the flow simply raisesthe pressure level everywhere by the same constant level, and does not alter the flow behavior.

Some Eact Solutions of the Navier-Stokes Equations

:e ne1t turn our attention to the e1act solutions of incompressible "avier-#to$esequations. Cven though these equations were derived over a century ago, only a handful of e1actsolutions e1ist for some highly simplified situations. This is because of the nonlinear nature ofthese equations.

The governing equations are+

ux

+vy

= 0

Du

Dt+

p

x= 2 u

DvDt

+ py

= 2 v

DE

Dt+

(up)x

+ vp( )

y=

uxx + vxy( )x

+ uxy + vyy( )

y+ k2T

where,

D

Dt=

t

+ ux

+ vy

E = CvT +u2 + v2

2

In the above equations, we have assumed the conductivity $ and viscosity to be constant. :e

will further restrict ourselves to steady flows (H6Ht 3 )! , and incompressible flows (3 constant!.

The above form of equations apply only to 2-D planar flows. #imilar forms of "avier-#to$es equations e1ist in other coordinate systems such as the cylindrical, polar and sphericalcoordinate systems.

Garallel lows

The first class of flows to be considered is the flow between (or within! infinitely longparallel plates, and infinitely long tubes.

1

y

1

r


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In such flows, it is reasonable to assume that the flow behavior will be independent of the1- location. That is, 1-derivatives of all flow properties such as Hu6H1, Hv6H1, HT6H1 etc. vanish. Thepressure derivative Hp6H1 is assumed to be a constant, required to drive the flow.

In such as situation, continuity equation becomes

Cartesian Form:ux

+vy

= 0

Cylindrical Coordinates :

ux

+1

r

rv( )r

= 0

#etting Hu6H13) yields Hv6Hy 3 ) or H(rv!6Hr3). The assumption Hv6H13) implies that thevelocity component v is not a function of 1.

Thus, continuity yields v 3 constant, for our flows. #ince the boundary condition requiresv to be Aero at the walls of our parallel plates, and at the walls of the tube, continuity @ boundaryconditions together yield

v =0

In other words, the flow only has a u- component and is parallel to the 1- a1is. or thisreason, the first class of flows we study are called parallel flows.

C1act #olution = %+ Glanar *ouette low+

In %&L) *ouette performed e1perimental studies of flow between two concentriccylinders, where one of the cylinder is fi1ed, the other is spinning. This situation is similar to what

happens in a ball bearing. here we study the 2-D analog of flow between two parallel plates. neof the plate is held fi1ed, while the other one is moving at a velocity ;. The plates are separatedby a distance h. constant pressure gradient dp6d1 is applied to this flow.

h

u3) at y3)

u3; at y3 h

dp6d13*onstant

v3 ) everywhere

1

y

or this case, the u- momentum equation yields


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uux

+ vuy

+ 1

dp

dx=

2ux2

+2uy2

where,

=

= Kinematic Viscosity

#etting the 1- derivatives to Aero, and requiring v to ve Aero, the above equation may besimplified to yield

1

dp

dx=

d2u

dy2

"otice that we have begun to replace partial derivatives with respect to y with ordinaryderivatives, because flow properties are only functions of y.

Integrating the above equation twice, we get+

u(y) =1

dp

dx

y2

2 +Ay +B

where and < are constants of integration. These may be evaluated by requiring u3) at y3) andu3 ; at y3h.

u(y) =Vy

h+

1

dp

dx

y

2y h( )

u(y) =Two special situations are of interest. In the first case, the pressure gradient dp6d13) is

Aero, and the flow motion is brought about by the motion of the top plate at the constant velocity;. In that case, the velocity profile is linear, giving

u(y! 3 ; y6h

In the second situation, both the plates are at rest and ;3). the fluid is motion is causedby the application of pressure gradient dp6d1. In this case, the velocity profile is a parabola, givenby

u(y) = 1

2dp

dxy y h( )

The shear stress 1y

(i.e. the force per unit area e1erted by the fluid on the flat plates, and

vice versa! may be computed from #to$esB relations. or e1ample, for the case where dp6d13),we get

xy = uy

+vx

=

V

h

The non-dimensional >s$in friction> coefficient *fis then computed as


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Cf =

xy

1

2V2

= 2hV

= 2Re

where,

Re = Reynolds number based on V and h =Vh

Temperature Distribution for Glanar *ouette low+

The energy equation, which is part of the "avier-#to$es equations is usually not solved inincompressible flow applications, unless we are interested in a heat transfer application. ?et usconsider the situation where dp6d13), the top plate is moving at the velocity ;, and the pressuregradient dp6d13). ?et the bottom and top plates be at two different temperatures T

% and T

2respectively. In this case, we can easily solve for the temperature distribution in the fluid betweenthe plates, and compute the rate at which heat is transferred from the hot plate to the cold plate,as follows.

u3 ; y6h

u3; T3 T2

u3) T 3 T%

dp6d13)h

The energy equation for this case is (;erify for yourself, starting with the form given in/andout =%!+

uh0( )x

+ vh 0( )

y=

uxx + vxy( )x

+ uxy + vyy( )

y+ k2T

where,

h0 = Stagnation enthalpy = CvT + p +u

2

+ v2

2

or our parallel flow, we can set H6H1 3) and v 3 ). or the case where dp6d13), thevelocity profile is linear, as derived earlier. Then,


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xx =2ux

+ ux

+vy

=0

yy

=2vy

+ ux

+vy

=0

xy

= uy

+ vx

= V

h

:hen these e1pressions are substituted into the energy equation, and when all 1-derivatives are set to Aero, and when v is set to Aero, the following form results.

k2Ty2

= V2

h2

Integrating the above equation twice, we get

T(y) =

2k

V2

h

2y2 +Cy +D

where * and D are constants of integration. These constants may be found by applying theboundary condition T3 T

%at y3), and T 3 T

2at y3 h. The final form is

T(y) = T1 +T2 T1

hy+

2k

V2 y

h

y2

h2

The first two terms in the above e1pression are linear in y, and model the effect ofconduction on the temperature distribution. The third term models the effect of heat geeration due

to viscous wor$. #ince is small, this term is li$ely to be of significance only in high speed flows.

ns equations

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