Chapter 6 An introduction of turbulent boundary layer T-S Leu May. 23, 2018 Chapter 6: An introduction of turbulent boundary layer Reading assignments: 1. White, F. M., Viscous fluid flow. McGraw-Hill, 1974, Chapter 6. 2. Klebanoff, P. S., Characteristics of turbulence in a boundary layer with zero pressure gradient, NACA Technical Note 3178, 1954. 3. Clauser, F. H., The turbulent boundary layer, Advances in Applied Mechanics, Vol. 4, pp. 1-51, 1956. 4. Clauser, F. H., Turbulent boundary layers in adverse pressure gradients. Journal of the Aeronautical Science, vol. 21, 1954, pp. 91-108. 5. An Album of Fluid Motion, ed. M. van Dyke, 1982. This chapter provides derivations of the equations relevant to turbulent flows, the measurement data of a flat-plate turbulent boundary layer, the concept of the equilibrium boundary layer, and the physical aspects concerning the structure of turbulent boundary layer. 2 1
Definitions of Turbulence -Taylor and von Kármán (the years of 1930) Turbulence is an irregular motion which in general makes its appearance in fluids, gaseous or liquid, when they flow past solid or even when neighboring streams of the same fluid flow past or over one another. G. K. Batchelor, Geoffrey Ingram Taylor, 7 March 1886-27 June 1975 JFM (1986), Vo1. 173, pp. 1-14. W. R. Sears, M. R. Sears, The Karman years at CALCIT, Annual Review of Fluid Mechanics, 1979, Vol. 11, pp.1-10. -Hinze (1959) Turbulent fluid motion is an irregular condition of flow in which the various quantities show a random variation with time and space coordinates, so that statistically distinct average values can be discerned. 3 Definitions of Turbulence (cont d) -Bradshaw (1971) Turbulence is a three-dimensional timedependent motion in which vortex stretching causes velocity fluctuations to spread to all wavelengths between a minimum determined by viscous forces and a maximum determined by the boundary conditions of the flow. 4 2
derstandings of turbulence are from complete yet! 5 The origin of turbulence The origin of turbulence can be linked to the process of laminarturbulent transition. Initially, flow instability develops when a threshold condition is reached. The initial flow instability appears to be very small in amplitude of velocity and pressure. Thus, the disturbances can be represented by some harmonic waves whose frequency is the most prominent one in the initial development phase. Later in time or space, the disturbances grow in amplitude, subsequently so large that the wave form is appeared distorted, inferring that the disturbances at this stage contain more than one frequency component. More frequency components will come in the later development of flow instabilities. Eventually, the flow disturbances are characterized by a wide band frequency spectrum. This state is referred to as turbulence. 6 3
Some examples of turbulent flows -A study of water formation by Leonardo da Vinci in 15 Centery 7 Multi-scales in turbulence -Eruption of volcano 8 4
-Turbulent boundarylayer flow (Van Dyke, 1982) 9 It is impossible to predict a turbulent flow at an instant of time rigorously by theoretical, numerical or experimental methods. Instead, there are a number of approximation methods from the theoretical, numerical and experimental points of view. However, since they are approximations, not exact rigorously speaking, these methods are only valid for certain classes of turbulent flows. Here, we are particularly focused on the turbulent boundary layer. It is impossible to solve the real-time (instantaneous) equations of a turbulent flow with the current technology, even the initial and boundary conditions are available. An alternate method is to adopt the Reynolds decomposition method. Reynolds decomposition separates a turbulence quantity into a mean and a fluctuating components. 10 5
Reynolds decomposition method (I) Let us take a look at why. Consider the two instantaneous quantities A and B which we decompose to A = Ā + A where the overline indicates a temporal average over the time scale of interest. Based on our definition of the Reynolds decomposition we have: and importantly the nonlinear product yields: where If A & B are random fluctuations (independent), then =0 is the correlation of fluctuating quantities. Reynolds decomposition method (II) 6
If mean flow is steady, two-dimensional and neglect body forces 13 Take time average over the equations 14 7
Reynolds Average Navier Stokes (RANS) equations in tensor form: Reynolds stress terms Shear stress Reynolds stress or turbulent stress Total stress Shear stress Reynolds stress or turbulent stress Further, introduce the 2D boundary layer approximation 16 8
There are no general solutions for RANS equations above. In fact, the number of unknowns in the equations are more than the number of the equations. Unless more relations, these equations can not be solved. On the other hand, it is interesting to have a discussion of the x-direction momentum equation in the following form. U u'v' y It is also interesting to note Pressure variations in turbulence boundary layer How large is it? 17 If a parcel sitting at the point (x o, y o ) has a fluctuating vertical velocity v > 0, it has high probability that u < 0 since it is low momentum fluid moving to a region of high momentum. Now consider hundreds and thousands of such parcels, on average we expect u' v' 0 18 9
Discussion of the momentum equations will not be carried on further, since it is no way to obtain the solutions of the equations without empirical modeling. Note that the desirable flow quantities are the velocity distribution and turbulence intensity profiles in boundary layers. Experimental observations of the mean velocity profiles of turbulent boundary layers For Reynolds number high enough, the mean velocity profile of a turbulent boundary layer consists of three regions, namely, the inner region, the log region and, the outer region. This can also be seen by the physical arguments in the next page. the outer region Klebanoff (1954), Figure 3 the log region the inner region 19 Shear Stress and Friction Velocity The shear stress (= rate of transport of momentum per unit area in the positive y direction) is U u'v' y Near the wall, inner region (say y/ <0.1): U u'v' y U w y The viscous part varies from being the sole transporter of momentum at the wall in the inner region of a turbulent boundary layer. Away from the wall, outer region (say y/ >0.3): U u'v' y u'v' The direct effect of viscosity is negligible. Reynolds stress being the sole transporter of momentum in the outer region of a turbulent boundary layer. 10
Length and Velocity Scales in Turbulent Boundary Layer Velocity Profle inner region: 1. Very close to the wall the most important scaling parameters are: kinematic viscosity & wall shear stress w 2. The characteristic velocity and length scales are: friction velocity: u* u w / Viscous length scale: v / u * 3. From these we can form non-dimensional velocity and height in inner region: U U / u * y y / v yu* / 4. y + is a sort of local Reynolds number. Its value is a measure of the relative importance of viscous and turbulent transport at different distances from outer region: the wall. 1. At large y+, the direct effect of viscosity on momentum transport is small and heights can be specified as a fraction of the boundary-layer depth : y / 2. The friction Reynolds number is a global parameter of the boundary layer and defined as: Re u* / Since turbulent boundary-layer flow is specified by U, y,,, and u *, dimensional analysis (6 variables, 3 independent dimensions) yields a functional relationship between 6 3 = 3 dimensionless groups, conveniently taken as U / u* f ( yu* /, y / ) i.e. U f ( y, ) All turbulent boundary-layer analysis is based upon the smooth overlap of the limiting cases inner layer ( 0) and outer layer (y+» 1). -The concept of inner and outer scales -Consider the turbulent motions in the inner region (0) where the viscous effect is dominant, the velocity and length scale should be scaled with the viscous effect. Friction velocity: Viscous length scale: Viscous sublayer (y + <5): u / y u* u w / 2 u / y / u / y w / / u u / u* y /( / u* ) y u * For y 5, u y 22 11
Experimental observations of the viscous sublayer (inner region) of turbulent boundary layer Viscous sublayer (y + <5): For y 5, u y the outer region the log region the inner region 23 -Consider the turbulent motions in the outer region, the length scale should be comparable to the thickness of the boundary layer. The velocity defect should be small, like the velocity in the downstream turbulent wake region behind a bluff body. Therefore, the velocity defect is scaled by the friction velocity. U U y F( ) u* the velocity in the downstream turbulent wake region behind a bluff body u u 24 12
- Logarithmic law Between the outer and inner regions, there is a region where the flow characteristics show a transition from one region to the other. This region is also regarded as an overlapping region, where can be described by either inner or outer scales. Consider the mean velocity in this region 25 26 13
Universal Velocity Profile or logarithmic velocity distribution law. Therefore, the velocity profile in this region, expressed in terms of is a logarithmic curve. 27 u* u w / 28 14
Experimental observations of the inner region, and the log region of turbulent boundary layer u y u u 2.5ln y u* 5.0 u u 5.6log10 y u* 5.0 29 15
Wind velocity profile of an atmospheric boundary layer The expression of the logarithmic profile (Log Law) U* U( z) ln( z / z0) K http://mae.engr.ucdavis.edu/~wind/facilities/blthickness.jpg Calm open sea: 0.20mm; Lawn grass: 8mm; Suburbs: 1500mm; centers of cities with tall buildings: 3000mm Wind Energy Explained Manwell, J. F., McGorwan, J. G., Rogers, A. L., Wind Energy Explained, 2nd edition, 31 John Wiley and Sons, 2009. Example: Wind velocities over open fields were measured as 5.89 m/ s and 8.83 m/s at heights of 2 m and 10 m respectively. Use this data to estimate: (a) the roughness length z0; (b) the friction velocity u * (c) the velocity at height 25 m U U * * U( z) ln( z / z0) U( 2) 5.89 ln(2 / z0) K K 5.89 ln(2 / z0) ln(2) ln( z0) 8.83 ln(10 / z ) ln(10) ln( z ) 0 U* U( 10) 8.83 ln(10 / z0) K 0 5.89 0.693 ln( z0) 8.83 2.302 ln( z ) 13.562 5.89 ln( z0) 6.119 8.835.89 ln( z0) 2.94 ln( z0) 7. 442 ( a) z 0 0. 08m U* ( b) 5.89 ln(2 / 0.08) U * m / s 0.4 0.73 0.73 0.73 ( c) U(25m) ln(25/ 0.08) U ( 25m) ln(25/ 0.08) 10.48m / s 0.4 0.4 0 16
Experimental Turbulent Boundary Layer Velocity Profiles for various pressure gradients (I): inner region -Experimental data of mean velocity profile Referring to Fig. 4 in Clauser (1954), the mean velocity profile near smooth walls can be plotted with the inner scales 34 17
The Clauser chart method In the experimental study of turbulent boundary layers, the determination of the friction velocity u*, is critical, since most of the scaling laws for the turbulent boundary layer involve u*. Unfortunately, data that come from direct measurement of the wall shear stress are not always available, requiring the use of indirect methods to deduce the wall shear stress. Boundary layer experimentalists commonly use the Clauser chart method which assumes a logarithmic law for the mean velocity profile. It is easy for experimental data fit to find skin friction and u *. - Clauser chart method (easy for experimental data fit to find skin friction and u * ) Referring to Fig. 5 in Clauser (1954), the mean velocity profile near smooth walls can be plotted with the Reynolds number 36 18
-Concept of the equilibrium turbulent boundary layer In earlier studies, a question was asked regarding whether there would be turbulent boundary layers under certain conditions bearing the similarity behavior like the laminar boundary layers of Falkner-Skan flows. Clauser (1954) illustrated that this class of turbulent boundary layers is achievable. Moreover, in the study he proposed a characteristic length to scale turbulent boundary layers. 19
Equilibrium turbulent boundary layer 20
Experimental Turbulent Boundary Layer Velocity Profiles for various pressure gradients: outer region 21
or 22
Referring to Fig. 5.17 in Tennekes and Lumley (1972) Tennekes and Lumley (1994) showed that when the non-dimensional parameter is constant, the boundary layer is self-preserving, like self-similar laminar boundary 45 layer. Such a class of t. b. l. are called the equilibrium turbulent boundary layers. Experimental data on the velocity-defect law (law of the wake) Referring to Fig.5.8 in Tennekes and Lumley (1972) for the case of pipe flow 23
-Experimental data concerning the characteristics of turbulent fluctuations in the boundary layer In wind tunnel experiment, turbulent fluctuations can be measured by hot-wire probe. The instantaneous velocity trace can be depicted as follows. 47 Turbulence intensity : Turbulent fluctuations are always three-dimensional. 48 24
-Referring to Fig. 4 in Klebanoff (1954) for the measurement data obtain in a flat-plate boundary layer It is always difficult to get measurements very near the wall. 49 A schematics of energy cascade in turbulence: Mean flow large scale eddies Small scale eddies Dissipation to Internal energy 2-D, 3-D flow instability Shear flow turbulence energy production Vortex stretching and tilting Vortex stretching and tilting Dynamically equilibrium Vortex stretching and tilting 50 25
Vorticity Equation Assumptions: density and viscosity is constnat Conservative body force G V 2 1 p ( VV) Vω G ( ) V t 2 V 1 p ( VV) Vω G ( ) 2 V t 2 (...) 0 ω ( Vω) 2 ω t ( V ω) ( ω) V( V) ω ω ω ( Vω) ( ω) V( V) ω 2 ω t t ω 2 ( V) ω( ω) V ω t Rate of change of particle vorticity Dω 2 ω( ω) V Dt Net rate of viscous diffsuion of vorticity f j Rate of deformation of vortex lines 26
-3-D vorticity fluctuations: turbulence is rotational and 3-D in nature. The mechanism of vortex stretching is prominent in the dynamics of turbulence. This mechanism plays the major role in the cascade of turbulent energy from large to small eddies. V Vorticity equation for incompressible flow Vorticity stretching and tilting Vorticity velocity 流體力學知多少, 苗君易, 國立成功大學,2003. 53 Dω 2 ω( ω) V Dt 54 27
Kelvin Theorem Vorticity intensification by stretching vortex lines Inviscid fluid (=0) G f Conservative body force j G x j Fluid density = constant D 0 Dt da or barotropic (p=g()) A A D( ) D ( A) D( ) D( A) A 0 Dt Dt Dt Dt D( A) D D( A) 0 if 0 / A Dt Dt Dt D( A) 0 if 0 Dt If a fluid particle is 2 1 1 1 1 A1 2 2A2 1 stretched in an inviscid flow field, its vorticity in the stretch direction is increased. The effect of vorticity stretching and tilting Physical speaking, the effect is to change the scales of the turbulence eddies, especially for the stretching effects. This can be argued from Kelven theorem Conservation of angular momentum of the eddy, 28
The case of stretchibg which leads to vorticity increase and the eddy size decrease is important to energy cascade because, as the scales get smaller, they dissipate to internal energy. The main path of energy cascade is from kinematic energy drained out to internal energy. A schematic of vortex stretching process and energy cascade in turbulence (Bradshaw, 1971) Turbulent eddies For a two-dimensional flow, no vorticity stretching and tilting effects. You may check the vorticity stretching and tilting term in the vorticity equation. 流體力學知多少, 苗君易, 國立成功大學,2003. 58 29
A global view of a turbulent motion Turbulent motion contains many eddies. Each eddies is 3D and time dependent. There are strong interaction between eddies Vortex stretching is occurred in the 3-D flow Vortex stretching leads to reduction of the length scale of turbulent eddies and increase of vorticity. (Law of conservation of angular momentum) The vortex stretching process prevails in a turbulent flow. The dynamic process is like a chain reaction. For instance, the vorticity stretching in one direction will further induce the vorticity stretching in the other two directions. By nature, a turbulent flow contains a wide range of length scales of eddies in various orientations. Each eddy in which rotational fluid contains vorticity affects and is affected by the surrounding eddies as the mechanism of vorticity stretching is at work. Through this mutual interaction process the length scales of turbulent eddies keep evolving, most likely get smaller and smaller toward the Kolmogrov scale. The smallest scales are called Kolmogrov scales, below which the turbulent fluctuations would not be possibly existed. By dimensional analysis, Kolmogrov length scale: corresponding velocity scale Kinematic viscosity Energy dissipation corresponding time scale 60 30
Overall speaking, turbulence is produced by velocity gradient of shear flow, so-called shear flow turbulence. Mean shear produces large scale fluctuations, whose scale is comparable to the dimension of the flow. Further, through the vorticity stretching, the energy spread to different scales of eddies. Meanwhile, the dissipative mechanism is at work. Mean flow provides kinetic energy to large scale eddies through the action of mean shear. This action can be seen in the turbulence kinetic energy equation, identified as the turbulent energy production term. Boundary layer mixing layer Mean shear Initial instability Large scale eddies 3-D turbulent fluctuations 61 Length scales in turbulent flows By nature, a turbulent flow contains fluctuations spreading over a wide frequency spectrum. Equivalently speaking, a turbulent flow contains different scales of eddies. It is a challenging issue to describe a turbulent flow with so many frequency and length scales. Usually, one has to identify the dominant scales associated with the turbulent flow of interest, which are referred to as the characteristic scales. The fewer the characteristic scales, the easier the description can be made. 62 31
-Consider a laminar boundary layer developed on the surface of an airfoil Chord length of the airfoil; convective length scale Boundary-layer thickness; Diffusive length scale convection diffusion 63 -Consider a turbulent boundary layer Introducing a new velocity scale u characterizing fluctuating velocity The growth of the turbulent boundary layer thickness is associated with the turbulent velocity fluctuations. Therefore, The characteristic velocity associated with large scale eddies Convection time scale Therefore, 64 32
-Compare frictions of laminar and turbulent boundary layers -laminar boundary layer case: Wake mom. deficit: Laminar boundary layer Drag: Drag coefficient: 65 -turbulent boundary layer case; Wake mom. deficit: Drag: Drag coefficient: Experimental observation 66 33
Experimental evidence shows that the turbulence level varies very slowly with Reynolds number, so that the drag coefficient of a turbulent boundary layer should be insensitive to Reynolds number. The drag coefficient of a flat plate 67 -Scales in turbulence characterizing the velocity scale of large scale eddies Large scale eddies do most of the transport of momentum and contaminants Large scale eddies are comparable to the characteristic length of the mean flow How to describe the length scales of small scale eddies? Why it is a necessity to look into the small scales of turbulence? What is the role of small scale eddies in turbulence dynamics? 68 34
Although turbulence is usually seen at high Reynolds number, this does not imply that the effect of viscosity is not important. On the contrary, the effect of viscosity plays the major role in energy dissipation, which is mainly undertaken by small scale eddies. The smaller the eddy size the more significant the energy dissipation. At high Reynolds number, the length scales of large and smallest eddies differ by several orders of magnitude. Hence, they are not directly related. As noted earlier, the smallest length scale of turbulent fluctuations is referred to the Kolmogrov length scale. 69 An inviscid estimate for the dissipate rate Energy supply from large scale turbulence Small scale eddies Energy dissipation Energy extracted from mean flow Kinetic energy per unit mass: When the turbulent flow is dynamically balanced, Time scale: Is equivalent to the energy dissipation rate 70 represents the length scale corresponding to the large scale eddies. 35
On the other hand, the energy dissipation rate of large scale turbulence can be estimated as follows. Dissipation is associated with viscous effect, therefore the viscous time scale of large eddies can be represented by Therefore, the dissipation rate can be estimated as A comparison of the dissipation rates of small and large scale eddies indicates 71 -Scale relations With the energy dissipation represented by The characteristic scales of large and smallest eddies can be shown below. Ratio of length scales: Ratio of time scales: Ratio of velocity scales: The higher the Reynolds number, the wider the range of the scales. 72 36
Energy Equations for Turbulent Flow Turbulent energy production term, due to the interaction of turbulent stresses and mean shear flow 37
Kinetic energy equation for turbulent fluctuating motion Rate of change of kinetic energy in turbulence fluctuations of a specific fluid particle The convective diffusion by turbulence of the total turbulence mechanical energy The work of deformation of the mean flow by the turbulence stresses, turbulence energy production The work by the viscous stresses of the turbulent motion Viscous dissipation by the turbulent motion 75 Turbulence energy production The work of deformation of the mean flow by the turbulence stresses, turbulence energy production term for 2D boundary layer: u j ui ' u j ' x i u u' v' y 38
Boundary layer control- drag reduction L. W. Reidy, Flat Plate Drag Reduction in a Water Tunnel Using Riblets, NOSC TR 1169, Naval Ocean Systems Center, 1987. Streamwise microgroove surface modifications, known as riblets, are known to alter the structure of the turbulent boundary layer such that the skin friction drag over the surface is reduced. For this investigation, velocity profile measurements were used to calculate the drag on a flat plate, both with and without riblets, in a high-speed water tuninel; The vinyl riblet surface, produced a maximum drag reduction of 8.1% ±2.5% at a non-dimensional groove spacing, h+=s + of 13.1. If the results obtained In the laboratory can be realised for torpedoes and other submersibles, significant-improvements in vehicle speed or fuel economy will be achieved. 77 http://rsta.royalsocietypublishing.org/content/368/1929/4775 39