Momentum Heat Mass TransferMHMT6

Solutions of the Navier-Stokes equations in limiting cases. Engineering Bernoulli equation. Boundary layer theory. Flow along a plate. Karman integral theorem.

Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010

Bernoulli equation. Boundary layer.

sourceDt

D

Navier Stokes - high ReMHMT6

2 2

2 2

1( )x x x x

x y

u u u upu u

x y x x y

Let us consider the 2D Navier Stokes equation describing for example a steady state fluid flow around a body (diameter D) at incoming velocity U. The following equation represents a momentum balance in the incoming flow direction and should have been used for prediction of drag forces

2 2

2 2 2

acceleration viscous forcespressure forces(friction drag)(form drag)

1 1( )

Rex x x x

x y

U U U UpU U

X Y U X X Y

Introducing dimensionless variables X=x/D, Y=y/D, Ux=ux/U, Uy=uy/U, Re=UD/ gives dimensionless equation

It seems to be obvious that with the increasing velocity (with increasing Re) the Navier Stokes equation reduces to the Euler’s equation and the last term of viscous forces is less and less important (all dimensionless variables X,Y,Ux,Uy are supposed to be of the unity order, with the exception of Re).

d’Alembert’s paradoxMHMT6

D’Alembert’s paradox. Analytical solutions based upon Euler equation indicate, that the resulting force (integrated along the whole surface of body) should be zero. It was a great challenge for the best brains of 19th century (Lord Rayleigh, Lord Kelvin, von Karman) to explain the controversy between experience (scientists knew about the quadratic increase of drag forces with velocity U) and theory represented either by the Stokes solution for drag on a sphere (cD=24/Re) or steady state solutions of Eulers’ equations predicting zero drag for high Reynolds number. Suspicion was focused to possible discontinuities/ instabilities of potential flows, and to wake, resulting to explanation of many important phenomena, for example the Karman vortex street (see previous lecture, or read von Karman paper, T.Karman: Über den Mechanismus des Wiederstandes, den ein bewegter Körper in einer Flussigkeit erfahrt. Nachrichte der K. Gesellschaft der Wissenschaften zu Göttingen Mathematisch-physikalische Klasse. (1911) 509-517).

These 7 pages caused similar revolution like the Einstein’s relativity theory. Prandl realised that the dimensional analysis can be misleading and that the last term on the right hand side of the previous equation cannot be neglected even for infinitely large Re, because viscous fluid sticks at wall and very large velocity gradients exist in a thin boundary layer. The whole flow field is to be separated to an inviscid region and to a boundary layer, described by parabolised NS equations.

Prandtl L.: Über Flussigkeitsbewegung bei sehr kleiner Reibung. Verhandlungen d.III Internat.Math.Kongress, Heidelberg 8.-13.August 1904, B.G.Teubner, Leipzig 1905, S.485-491

Quite different view was suggested by Ludwig Prandtl during conference in 1904:

Boundary layerMHMT6

Outside the boundary layer the velocity field is described by the Euler equation. For inviscid incompressible flow the relationship between velocities and pressure are expressed by Bernoulli theorem

xy

UH

pU

2

2

1

Boundary layer MHMT6

Inspection analysis must be carried out with a great care taking into account relative magnitude of individual terms in the Navier Stokes equations (and to distinguish magnitudes in longitudinal (1) and transversal () direction).

….terms of the order can be neglected

(replace x=L, y=, ux=U, uy=U, =2)

2 2

2 2

1( )x x x x

x y

u u u upu u

x y x x y

2 2

2 2

1( )y y y y

x y

u u u upu u

x y y x y

0yxuu

x y

U2/L(1) U2/(1) 2U/L2(2) U(1)

U2/L() U22/() 3U/L2(3) U ()

U/L(1) U/(1)

Boundary layer MHMT6

Equations describing boundary layer are reduced to

2

2

1x x xx y

u u upu u

x y x y

10

p

y

0yxuu

x y

The last equation shows that the pressure is constant in the transversal direction and its value is determined by Bernoulli equation applied to outer (inviscid) region,

Hp

U

2

2

1

2

2x x x

x y

u u uUu u U

x y x y

therefore the momentum equation in the x-direction is

Remark: While the original Navier Stokes equations are elliptic, the boundary layer equations are parabolic, which means that we can describe an evolution of boundary layer in the direction x (this coordinate plays a similar role as the time coordinate in the time evolution problems). This is a great advantage because marching technique enables step by step solution and it is not necessary to solve the whole problem at once.

Boundary layer - separationMHMT6

These equations are the basis of Prandtl’s boundary layer theory, which offers the following two important results:

1. Viscous (friction) drag forces, called skin drag can be predicted. As soon as there is no separation of boundary layer from the surface the outer flow is not affected by the presence of boundary layer. The outer flow can be therefore solved in advance separately giving (via Bernoulli’s equation) pressure and boundary conditions for the boundary layer region.

2. Form drag. Point of separation of the boundary layer on highly curved surfaces (cylinders, spheres, airfoils at high attack angles) can be predicted too. The separation occurs at the point with adverse pressure gradient, and the separated boundary layer forms a Helmholtz discontinuity surface. Behind the discontinuity is formed a wake (“dead fluid” region) increasing the form drag (drag caused by pressure imbalances) significantly.

Boundary layer - Plate MHMT6

Probably the most important question is the thickness of boundary layer (x). Only qualitative answer follows from the boundary layer equation (parallel flow along a plate with zero pressure gradient)

2

2x x x

x y

u u uu u

x y y

approximated very roughly as

2x x

x

u uux

giving ( )x

xx

u

x

This preliminary conclusion is qualitatively correct: boundary layer thickness increases with square root of distance, kinematic viscosity and decreases with increasing free stream velocity. See also the theory of PENETRATION DEPTH.

Boundary layer - PlateMHMT6

Little bit more precise analysis is based upon LINEAR velocity profile across the boundary layer. Integral balances of a rectangular control volume (height H)

Continuity equation

x

x+dx

+d

H

uy

U U

UdUdxudsun y

S

2

1 0

Integral balance in the x-direction

1 0x x

S S

n uu ds n ds

22 2

0 0 0

=- | | |6

H H dx

x x x x x dx y y H

S

U dn uu ds u dy u dy u U dx

=x

S

Un ds dx

U

xxd

Udx

12)(

6

Boundary layer - PlateMHMT6

Linear velocity profile results in not very precise solution ( ) 3.5x

xU

Cubic velocity profile is much better giving ( ) 4.64x

xU

Exact formulation of differential equation of boundary layer was presented by Blasius (Prandtl’s student), however this ordinary differential equation requires numerical solution.

Anyway, knowing approximations of velocity profiles, it is possible to calculate viscous stresses upon the plate and therefore the drag force. For linear velocity profile

xUWx

dxWUdxWWxUcF

xx

wD3

00

2 57.0)(2

1

1.14

ReD

x

c 1.328

ReD

x

c RexU x

linear profilemore acccurate Blasius solution

Von Karman Integral theorem MHMT6

Hopper

Prandtl equations of boundary layer are partial differential equations. Karman theorem derives from these equations ordinary differential equation, suitable for approximate solutions.

Ordinary differential equation is obtained by integration of continuity equation across the boundary layer (in this way transversal velocity is expressed in terms of longitudinal velocity) and by integration of momentum equation in the transversal direction across the whole boundary layer.

Integral equations MHMT6

xy

U

0

( ) 0yxuudy

x y

2

20 0

( )x x xx y

u u uUu u U dy dy

x y x y

0

( )y

xy

uu y dy

x

0

( )x x wx y

u u Uu u U dy

x y x

Continuity equation integrated across the boundary layer

Momentum equation

Elimination of transversal component using continuity equation

0

0 0 0 0 0 0 0

( ) [ ]y y

x x x x x x xy x x x

u u u u u u uu dy dy dy u dy u dy U dy u dy

y x y x x x x

0

(2 )x x wx

u u Uu U U dy

x x x

Integral equations MHMT6

Integrated momentum equation can be rearranged to

U

*

2

0

( )x x wx

u Uu U Uu U dy

x x x x

2

0 0

[ (1 ) ] (1 )x x x wu u ud dUU dy U dy

dx U U dx U

Momentum thickness **

Displacement thickness *

Displacement of surface corresponding to the same flowrate of ideal fluid

In a similar way the energy integral equation can be derived (multiply momentum equation by ux and integrate)

Energy thickness ***

3 2

0 0

2[ (1 ( ) ) ]x x xu u udU dy dy

dx U U y

Dissipation integral

Integral equations MHMT6

Integral momentum equation holds for laminar as well as turbulent flow

The only one differential equation is not enough for solution of 3 variables: displacement thickness, momentum thickness and shear stress. Approximate solutions are based upon assumption of similarity of velocity profiles in the boundary layer.

2 ** *( ) wd dUU U

dx dx

Integral equations MHMT6

The simplest example is laminar boundary layer at parallel flow along a plate. In this case U=constant and integral momentum balance reduces to

**

2wd

dx U

Better approximation than the previously analyzed linear velocity profile ux=Uy/ is a cubic velocity profile, because the cubic polynomial with 4 coefficients can respect 3 necessary boundary conditions corresponding to laminar flow: zero velocity at surface y=0, prescribed velocity U and zero stress dux/dy at y=.

33 1( )

2 2xu y y

U

Integral equations MHMT6

Substituting this velocity profile to definition of ** and w=dux/dy

Karman integral balance reduces to the ordinary differential equation for thickness of boundary layer

**

0

39(1 )

280x xu u

dyU U

3

2w

U

39 3

280 2

d

dx U

Solution is the previously presented (but not derived) result

280 4.64

13 Rexx Ux

Please mention the fact that the momentum thickness ** is much less than the boundary layer thickness

Integral equations MHMT6

**

2wd

dx U

Turbulent boundary layer is described by the same integral equation, but it is not possible to use the same velocity profile (this is not true that duz/dy is zero at y=) and first of all the turbulent wall shear stress cannot be expressed in the same way like in laminar layer.

Brutal simplification based upon linear velocity profile, and simplified Prandtl’s model of turbulence (=2U2) gives linearly increasing boundary layer thickness

2 2w U 26

d

dx

In reality increases more slowly, with the exponent 0.8 ( ) – this prediction is based upon Blasius formula for friction factor (see textbook Sestak et al: Přenos hybnosti a tepla (1988), p.94). More accurate result (see Schlichting, Gersten: Boundary layer theory, Springer, 8 th edition 2000) is

8.0)( xx

**

0 0

(1 ) (1 )6

x xu u y ydy dy

U U

Re0.14 (ln Re )

ln Rex

xx

UG

G only weakly depends on Re and limiting value is

1

0.00E+00

5.00E+04

1.00E+05

1.00E+05 1.00E+06 1.00E+07

Integral equations MHMT6

Typical values of boundary layer thickness and thickness of laminar sublayer for plate (remark: even in the turbulent flow there is always a thin laminar layer adjacent to wall, see next lecture)

L

(L)U

Fluid U[m/s] L [m] Re [mm] lam [mm]

air 10 1

air 50 1 3.3106 8 0.4

water 1 2 2106 17 1

water 2 5 107 39 0.6

EXAMMHMT6

Boundary layers

What is important (at least for exam)MHMT6

Hp

U

2

2

1

2

2x x x

x y

u u uUu u U

x y x y

Bernoulli’s equation (outer inviscid region)

Boundary layer equation in the direction of flow

What is important (at least for exam)MHMT6

Thickness of boundary layer (plate)

Drag of parallel flow on plate

( ) 4.6x

xU

21

2 DF c U Wx 1.328

ReD

x

c RexU x

x