Chapter 1 Introduction of Boundary Layer Phenomena Main Topics

Chapter 1 Introduction of boundary layer phenomena

T-S Leu Jan. 13, 2018

Main Topics

• History of Fluid Mechanics Development • Idea of Boundary Layer • Boundary Layer Equations

EFD: Ideal fluid: Real fluid : Hot wire MEMS Inviscid flow (zero drag) Hot film Viscosity PIV 2000s Lagrangian/Eulerian Viscous flow Boundary LDV MD Potential flow Laminar/ layer 1990s 1980s DNS (Inviscid+Irrotational) Turbulent flow Ludwig 1960~70s CFD Osborne Prandtl Panel Turbulence Reynolds 1904 method modeling 1842~1912 Lagrange Richard (1827~1845) (1736~1813) 1842~1919 Feynman has Navier Laplace Lord described Cauchy Jean (1749~1827) Rayleigh turbulence as Poisson D'Alembert Hydrodynamic the most St. Venant Isaac 1752 1755 instability: important Stokes Newton 1738 Leonhard Taylor Rayleigh unsolved 1687 Daniel Euler Navier- Kelvin Helmholtz problem of Bernoulli Stokes Benard cells classical physics. 89y Equation

Bernoulli Equation

2 Bernoulli equation (I)

BERNOULLI’S EQUATION (II)

integration

BERNOULLI’S EQUATION

AERODYNAMICS (W2-1-2)

3 BERNOULLI’S EQUATION FOR AN IRROTATION FLOW

incompressibl e

Steady

Irrotational

No gravity

AERODYNAMICS (W2-1-2.1)

Unsteady Bernoulli Equation

This is not a very useful result in general since ∂vs/∂t can change dramatically from one point to another; to use this in practice we need to be able to draw streamline shapes at each instant in time. It works especially for simple cases such as impulsively started confined flows where streamlines have the same shape at each instant and we are interested in time required to start the flow.

4 Exercise 1 Flow out of a long pipe connected to a large reservoir,

(1)find the steady state velocity v2 in the pipe after the the transient stage

(2)find the transient velocity v2 in the pipe changing with time during the transient stage

Back to History

D'Alembert's paradox •Influid dynamics, d'Alembert's paradox is a contradiction reached in 1752 by French mathematician Jean le Rond d'Alembert. • D'Alembert proved that for incompressible and inviscid potential flow –thedrag force is zero on a body moving with constant velocity relative to the fluid. • Zero drag is in direct contradiction to the observation of substantial drag on bodies moving relative to fluids, such as air and Jean le Rond d'Alembert (1717- water; especially at high velocities 1783) corresponding with high Reynolds numbers.

https://en.wikipedia.org/wiki/D%27Alembert%27s_paradox Back to History

5 Lagrangian/Eulerian Description in Fluid Mechanics

Assignment: Watch the video about Lagrangian/Eulerian Description in Fluid Mechanics https://www.youtube.com/watch?v=mdN8OOkx2ko (MP4)

METHODS OF DESCRIPTION

• Lagrangian description => System • Eulerian description => Control volume

Ch 1-2

6 Lagrangian Description • Attention is focused on a material volume (MV) and follow individual fluid particle as it move. • The fluid particle is colored, tagged or identified. • Determining how the fluid properties associated with the particle change as a function of time.  Example: one attaches the temperature-measuring device to a particular fluid particle A and record that particle’s temperature as it moves about.

TA = TA(t)=T (xo,yo,zo, t)

where particle A passed through coordinate (xo,yo,zo) at to  The use of may such measuring devices moving with various fluid particles would provide the temperature of these fluid particles as a function of time. Ch 1-2

Eulerian Description

• Attention is focused on the fluid passing through a control volume (CV) fixed in the space. • Obtaining information about the flow in terms of what happens at the fixed points in space as the fluid flows past those points. • The fluid motion is given by completely prescribing the necessary properties as a functions of space and time.  Example: one attaches the temperature-measuring device to a particular point (x,y,z) and record the temperature at that point as a function of time. T = T ( x , y , z , t ) => field concept. The independent variables are the spatial coordinates ( x , y , z) and time t Ch 1-2

7 Field Representation of flow

• At a given instant in time, any fluid property ( such as density, pressure, velocity, and acceleration) can be described as a functions of the fluid’s location. This representation of fluid parameters as functions of the spatial coordinates is termed a field representation of flow. • The specific field representation may be different at different times, so that to describe a fluid flow we must determine the various parameter not only as functions of the spatial coordinates but also as a function of time. • EXAMPLE: Temperature field T = T ( x , y , z , t ) • EXAMPLE: Velocity field

V  u(x, y, z,t) i  v(x, y, z,t) j  w(x, y, z,t) k

Ch 1-2

Nature and Transformation of Lagrangian and Eulerian Description

• It is more nature to apply conservation laws by using Lagrangian description (ie. Material Volume). • However, the Eulerian description (ie. Control Volume) is preferred for solving most of problem in fluid mechanics. • The two descriptions are related and there is a transformation formula called material, total or substantial derivative between Lagrangian and Eulerian descriptions.

Ch 1-2

8 Material Derivative (I) • Let (x,y,z,t) be any field variable, e.g., ρ, T, V=(u,v,w), etc. (Eulerian description) • Observe a fluid particle for a time period t as it flows (Langrangian description) • During the time period, the position of the fluid particle will change by amounts x , y , z, while its vale of  will change by an amount       t  x  y  z t x y z x y z • As one follow the fluid particle, ( , , )  (u,v, w) So t t t D    x  y  z      lim       u  v  w Dt t0 t t x t y t z t t x y z which is called the material, total, or substantial derivative. Ch 1-3

Material Derivative (II)

• Use the notation D/Dt to emphasize that the material derivative is the rate of change seen by an observer “following the fluid.” • The material derivateive express a Langrangian derivative in terms of Eulerian derivatives. • In vector form, D        u  v  w   (V ) Dt t x y z t • May also use the “index notation” and Einstein’s “summation convention” (i.e, summing over repeated indices) to write D     uk Dt t xk

Where (x1,x2,x3)≡(x,y,z) and (u1,u2,u3)≡(u,v,w)

Note: The repeated index that us summed over is called a “dummy index”; one that is not summed is called a “free index”.

Back to History

9 Potential Flow Theory

Inviscid & Irrotational flow

GOVERNING EQUATION FOR IRROTATIONAL, INCOMPRESSIBLE FLOW: LAPLACE’S EQUATION

Continuity equation

Incompressible: constant For incompressible flow: there exists a streamfunction 

For irrotational flow: there exists a velocity potential Laplace’s equation  y= 0 For irrotational, incompressible flow, there are velocity potential and For irrotational, incompressible streamfunction that both satisfy flow: y Laplace’s equation. 2 2 2    0 x2 y2 2 2 2    0 Laplace’s equation x2 y2 AERODYNAMICS (W2_1_6)

10 Potential Flow Theory • Governing equation for Potential flow is Laplace equation 2 2 2    0 x2 y2 • Laplace’s equation is a second-order linear partial

differential equation. If 1, 2, 3, … , n represent n separate solutions of Laplace’s equation, then

=1+2+3+… +n is also a solution of Laplace’s equation.

• Complex potential with conformal mapping

Boundary Condition for LAPLACE’S EQUATION

Boundary Conditions: Infinite boundary conditions:

Wall boundary conditions:

or or

11 Fluid Flow Governing Equations

• Mass conservation => Continuity equation • Momentum equation F=ma=d(mV) /dt => Navier Stokes Equation •1st Thermaldynamic law (Conservation of Energy) => Energy equation

Fluid Flow Governing Equations • Continuity equation:

  D uk  (uk )     0 t xk Dt xk • Momentum (N-S) equation:

u u p  u    u u  j j k   i j  [ uk ]   ( )   f j t x x x x x x x  k j j k i   j i  • Energy equation: e e uk  T (  uk )   p  (k )   t xk xk xi x j u  u u  u   ( k )2   i  j  j   xk  x j xi  xi

12 2 Navier-Stokes Equations     Full N-S equation 3 u u p  u    u u  j j k   i j  [ uk ]   ( )   f j t x x x x x x x  k j j k i   j i   u For incompressible flows,   V  k  0 xk u u p    u u  j j   i j  [ uk ]     f j t x x x x x  k j i   j i  For incompressible flows with constant viscosity , u u p    u 2u  p 2u  j j   i j   j  [ uk ]    f j    f j t x x x x 2  x  2  k j   j i xi  j  xi  For incompressible ,inviscid fluids =0,

uj uj p [ uk ]  f j The Euler equations t xk x j

e e uk  T (  uk )   p  (k )   t xk xk x j x j u  u u  u   ( k )2   i  j  j e  C T   v xk  x j xi  xi T T u  2T k for constant C & k Cv (  uk )   p  k 2   v t xk xk x j 2 T T  T for incompressible flow Cv (  uk )  k 2   with constant Cv & k t xk x j

2 T T uk  T for inviscid flow with Cv (  uk )   p  k 2 constant Cv & k t xk xk x j T T 2T for incompressible inviscid Cv (  uk )  k 2 flow with constant C & k t xk x j v

13 where

where

14 Molecular and Statistical Approaches

• Fluids consist of molecules whose motion is governing by the law of dynamics. • The macroscopic phenomena are assume to arise from the molecular motion of the molecules. • The theory attempts to predict the macroscopic behavior of the fluid from the laws of mechanics and probability (or statistical) theory. Statistics => the predicted macro fluid behavior near an equilibrium state • For a fluid state not far from equilibrium, the molecular and statistical approaches yield the transport coefficients (such as the viscosity coefficient and the thermal conductivity), and the equations of mass, momentum and energy conservation. • The theory is well developed for light gases, but it is incomplete for polyatomic gas molecules and for liquids. Currie Ch 1-1

The idea of Boundary Layer The occurrence of the paradox is due to the neglected effects of viscosity. In conjunction with scientific experiments, there were huge advances in the theory of viscous fluid friction during the 19th century. With respect to the paradox, this culminated in the discovery and description of thin boundary layers by Ludwig Prandtl in 1904. (see the article: Ludwig Prandtl’s boundary layer, Physics Today, 2005, 58, no.12, 42-48).

Ch 1-0

15 • Prandtl made the hypothesis that the viscous effects are important in thin layers – called boundary layers – adjacent to solid boundaries, and that viscosity has no role of importance outside. •Theboundary-layer thickness becomes smaller when the viscosity reduces. The full problem of viscous flow, described by the non- linear Navier–Stokes equations.

2 ui ui p  ui  uj  Fi  t xj xi xjxj • Using his hypothesis (and backed up by experiments) Prandtl was able to derive an approximate model for the flow inside the boundary layer, called boundary-layer theory; while the flow outside the boundary layer could be treated using inviscid flow theory.

The principal concept of the boundary originally springs from the particular form of the fluid continuum equations in which the dissipation terms involve higher order derivatives than the inertial, advective terms, e.g. for the Navier Stokes equations for a non rotating fluid:

2 ui ui p  ui  uj  Fi  t xj xi xjxj For fluids like air or water the coefficient of viscosity  is often sufficiently small, in a non-dimensional sense to be clarified more formally below, such that the physical effects of friction would seem to be negligible allowing the neglect of the last term on the right hand side of the equation.

16 17 Pressure distribution of flow over a circular cylinder

Pressure distribution for the flow around a circular cylinder. The dashed blue line is the pressure distribution according to potential flow theory, resulting in d'Alembert's paradox. The solid blue line is the mean pressure distribution as found in experiments at high Reynolds numbers. The pressure is the radial distance from the cylinder surface; a positive pressure (overpressure) is inside the cylinder, towards the centre, while a negative pressure (underpressure) is drawn outside the cylinder.

https://en.wikipedia.org/wiki/D%27Alembert%27s_paradox

18 19 20 Reynolds’ experiment using water in a pipe to study transition from laminar to turbulence Laminar pipe flow Re < 2100

Transient pipe flow 2100

Turbulent pipe flow Re>4000.

Boundary Layer described by using vorticity source and vorticity sink point of view

21       ndA  inidA Stokes Theorem A A Consider a 2D uniform flow passing through a flat plate, boundary layer= is developing along x direction near the surface. All areas for = region= abcd are 1 with the length (a) Please calculate the circulation along the path abcd and the total vorticity inside region abcd at three locations 1, 2 and 3? Compare the total vorticities inside the region abcd at location 1, 2 and 3. (4%) (b) We know all vorciticity is generated on the boundary of surface. Is any vorticity generated on the boundary of surface from location 1 to location 2? If your answer is no, please describe the reasoning. If your answer is yes, please describe the location where the vorticity comes from. (4%) (c) Is any vorticity generated on the boundary of surface from location 2 to location 3? Why or why not? (4%) U U U bc bcbc Y 123 a d a d a d X

Boundary Layer • The German physicist Ludwig Prandtl suggested in 1904 that the effects of a thin viscous boundary layer possibly could be the source of substantial drag. • Prandtl put forward the idea that, at high velocities and high Reynolds numbers, a no- slip boundary condition causes a strong variation of the flow speeds over a thin layer near the wall of the body. This leads to the generation of vorticity and viscous dissipation of kinetic energy in the boundary layer. • Boundary-layer theory is amenable to the method of matched asymptotic expansions for deriving approximate solutions. In the simplest case of a flat plate parallel to the incoming flow, boundary-layer theory results in (friction) drag, whereas all inviscid flow theories will predict zero drag. • Importantly for aeronautics, Prandtl's theory can be applied directly to streamlined bodies like airfoils where, in addition to surface-friction drag, there is also form drag. Form drag is due to the effect of the boundary layer and thin wake on the pressure distribution around the airfoil.

22 -Growth of boundary layer due to the viscous effect (from the viewpoint of vorticity dynamics) •Non-slip boundary condition: no slip on a wall surface due to viscous effect (Prandtl, 1904). How does a boundary layer develop on a wall? •Consider flow over the leading edge of an airfoil, how the boundary layer is initiated and developed? (Lighthill, 1963)

In the neighborhood of the stagnation point, the external flow velocity U rises from its value zero at the first point of attachment to the maximum positive value, , therefore the vorticity-flow increases. Downstream of , , the vorticity-flow decreases, run into the possibility of flow separation. (Lighthill, 1963)

acceleration deceleration 46

23 Check a case

The concept of vorticity flux

24 -See the vorticity source and sink in the momentum equations.

Adverse pressure gradient; wall as a vorticity sink

Favorable pressure gradient; wall as a vorticity source

25 Boundary Layer Equations

-Boundary layer assumption and boundary layer equations ( White, F. M., Viscous fluid flow. McGraw-Hill, 1974. Chapter 4) •Characteristic scales of a boundary layer In the previous discussion, it was argued by intuition that near the wall

This relation can be further verified with the boundary layer assumption, which will be introduced in this section. First of all, let’s define the characteristic scales of a boundary layer as follows.

Airfoil surface 52

26 :characteristic length along the streamwise direction

:boundary layer thickness

:freestream velocity

Boundary layer assumption: (Nearly parallel flow assumption) This assumption implies that the boundary layer containing vorticity is relatively thin compared to the characteristic scale of a body, on which the boundary layer is developed. Therefore, the boundary layer is also referred to as a thin shear layer. In a broad sense, boundary layer is a term referred to all the shear layers of which the assumption is valid, including jet, wake and mixing layer.

•Non-dimensionalized conservation equations for two-dimensional, incompressible flows Continuity equation:

27 Momentum equations:

28 In a boundary layer, both of the convective and viscous diffusion effects should be considered. Therefore,

Reynolds number 57

Consequently, momentum equation in the x direction

Evaluate each term in the momentum equation in the y direction

2 2 2 2 l V0 U l  V0 U V0 U O( 2 )( )( ) ~ O( 2 )( )( )( ) ~ O( )( )  U U l  Ul U l U l

Reynolds number

29 Therefore,

Compared to the momentum equation in the x direction, the momentum equation in the y direction can be ignored.

This implies that pressure variation along the y direction, across the boundary layer, is insignificant.

The momentum equation in the y direction reduced to:

This implies that pressure outside the boundary layer of an airfoil surface can be evaluated by using potential flow results if separation flow (or stall) does not occur on the airfoil..

30 The boundary layer equation is referred to the momentum equation in the x direction.

Reynolds number

  D uk  (uk )     0 t xk Dt xk

uk Incompressible  0 xk

• The boundary-layer assumption is applicable when the Reynolds number is large. Cases of boundary layer phenomena: 1. wall-bounded shear layer 2. free shear layer: jet, wake and mixing layer

31 -Turbulent boundary- layer flow

M. Van Dyke, (ed.) An Album of fluid motion, The Parabolic Press, Stanford, California, 1982.

32 -Jet flow at low Reynolds number Re=2300 M. Van Dyke, (ed.) An Album of fluid motion, The Parabolic Press, Stanford, California, 1982.

-Comparison of turbulent wakes at high and low Reynolds numbers

M. Van Dyke, (ed.) An Album of fluid motion, The Parabolic Press, Stanford, California, 1982.

33 Flow Separation and Reattachment

-Flow separation and reattachment

Physically speaking, as flow over an object flow detached from the surface of the object is referred to as an occurrence of flow separation. On the other hand, a separated flow attached on the surface of an object is referred to an occurrence of flow reattachment.

• Sharp-edge separation: flow over a knife edge; flow over a delta wing

Flow separation insensitive to Reynolds number

• Boundary-layer separation: flow in a diffuser ; flow over an airfoil

34 Flow over a delta wing

Vortex breakdown

Vortex asymmetric breakdown

Flow separation in a diffuser

flow separation over an airfoil

35 -Two-dimensional boundary-layer separation

Prantl criterion

-Three-dimensional separation (Lighthill, 1963; White, 1974, p. 365, Fig. 4-40): Spiral focus ; separation line

71 Limiting streamline: streamline of flow very near the wall or the shear stress line

-Two-dimensional flow reattachment

Flow over a backward facing step

-Three-dimensional flow reattachment

Limiting streamline: streamline of flow very near the wall or the shear stress line 72

36 -Three-dimensional boundary layer: flows over 3-D surfaces

Flow over a swept-back wing ;atmospheric boundary layer

Note that flow near the wall is dominated by the pressure gradient

The velocity in the boundary layer changes direction with wall distance, 73 but is nearly parallel to the wall.

37 75

38 Vortex Breakdown

39 -Compressible boundary layer (White, 1974)

•Effect of Mach number •Shock-boundary layer interaction; shock induced separation

•Aerodynamic heating