Engineering, 2010, 2, 190-196 doi:10.4236/eng.2010.23027 Published Online March 2010 (http://www.SciRP.org/journal/eng/) The Effect of Near-Wall Vortices on Wall Shear Stress in Turbulent Boundary Layers Shuangxi Guo1, Wanping Li2 1School of Civil Engineering and Mechanics, Huazhong University of Science and Technology, Wuhan, China 2The Key Laboratory of Mechanics on Western Disaster and Environment of the Ministry of Education of China, Lanzhou, China Email: [email protected] Received June 10, 2009; revised July 9, 2009; accepted July 21, 2009 Abstract The objective of the present study is to explore the relation between the near-wall vortices and the shear stress on the wall in two-dimensional channel flows. A direct numerical simulation of an incompressible two-dimensional turbulent channel flow is performed with spectral method and the results are used to exam- ine the relation between wall shear stress and near-wall vortices. The two-point correlation results indicate that the wall shear stress is associated with the vortices near the wall and the maximum correlation-value lo- cation of the near-wall vortices is obtained. The analysis of the instantaneous diagrams of fluctuation veloc- ity vectors provides a further expression for the above conclusions. The results of this research provide a useful supplement for the control of turbulent boundary layers. Keywords: Spectral Methods, Two-Dimensional Turbulence, Wall Shear Stress, Two-Point Correlation 1. Introduction changed the near-wall streamwise vortices and achieved the purpose of control of the wall shear stress [6-8]. The flow phenomenon of turbulent boundary layers is Though the detailed mechanism has not been completely common in nature. It is closely related to aerospace, ma- clear so far, the above-mentioned control methods have rine, environmental energy, chemical engineering and made good effectiveness on wall shear stress reduction. other fields. In aeronautical engineering, the complex Recently Y. S. Park et al also researched the control of turbulent vortex structures in boundary layers not only wall shear stress by the method of wall blowing and suc- affect the working stability and security of the aircraft tion. Instead of vertical to the wall, certain angles were but also increase the skin friction on the wall remarkably. presented between the blowing-suction direction and the So the research of control of turbulent boundary layers is streamwise direction. This meant that the velocity significant. In recent years the related literatures focus boundary conditions brought by their control method mainly on two aspects: control of near-wall turbulent were normal and streamwise velocities instead of normal structures and wall skin friction. Essentially, the wall and spanwise velocities. Their experiments showed a skin friction is closely related with the near-wall turbu- better effectiveness of wall shear stress reduction if the lent structures, so the researches on these two aspects are angle was proper [9]. In fact, wall normal and stream- in accordance. In flat wall flows, the wall shear stress wise velocity boundary conditions changed the near-wall constitutes wall skin friction. Sheng, Malkiel and Katz spanwise vortices directly. As well as streamwise vor- did much in-depth and complete study on the relation tices, spanwise vortices are also the main characteristics between wall shear stress (streamwise and spanwise) and in turbulent boundary layers, while are they also the cru- near-wall flow structure (streamwise, spanwise and out- cial effect factors on wall shear stress as the streamwise side structure) by experiment method [1]. Most re- vortices? searchers agreed that the near-wall streamwise vortices Essentially, turbulent flow is absolutely three-dimen- were the main effect factors of wall shear stress [2-5]. sional, but some certain turbulence motion, such as at- The wall normal and spanwise velocities boundary con- mosphere or ocean flows, is behaving quasi-two-dimen- ditions were presented by the methods of wall blowing sionally. The horizontal scales are hundreds of kilome- and suction or spanwise-wall oscillation, which directly ters in the ocean and thousands of kilometers in the at- Copyright © 2010 SciRes. ENG S. X. Guo ET AL. 191 mosphere, while their vertical scale is only a few kilo- turbulence. Spectral method is one of the most common meters. So the turbulent motions in the vertical direction methods for the DNS of turbulence, which has many are suppressed and can be ignored can be treated as two- advantage, such as high degree of accuracy, quickly dimensional turbulence [10-11]. The saturation states of speed on convergence, and analytically spatial derivation two-dimensional turbulence have the similar characteris- for flow variables [13]. Many researchers have done lots tics as the three-dimensional turbulence, such as injec- of pioneering and significant achievements for this tion, sweep and other bursting phenomenon [12-13], method, such as John Kim, Moin & Moser [14], Kleiser while they have lots of differences from the three-di- & Schumann [15], Hu, Morfey & Sandham [16]. So mensional turbulence, such as self-organization and in- spectral method is applied to solve the Navier-Stokes verse energy cascade [10-11]. In addition, the simulation equation directly in the present study. of two-dimensional turbulence requires less expensive The governing equations for two-dimensional incom- computational resources in comparison with that of pressible channel flow can be written as the following three-dimensional turbulence. So it is also valuable for forms: the study of two-dimensional turbulence. Moreover, sca- lar vorticity in two-dimensional turbulence is controlled u p 1 i f 2u (1) by the normal and streamwise velocity, and has the same ii txi Re expression as the spanwise vorticity in three-dimensional vu ui turbulence, . So in the present paper the 0 (2) x y xi two-dimensional scalar vorticity is taken as the major Here, all variables are non-dimensionalized by the subject of study instead of the three-dimensional span- channel half-width and laminar Poiseuille flow cen- wise vorticity. The objective of the present study is to tral velocity U ; Non-linear term f includes the con- explore the relation between the near-wall vortices and c i the shear stress on the wall in two-dimensional channel vective terms and the mean pressure gradient [14]; and flows. Re denotes the Reynolds number defined as Re Uc / , So far, the researches of two-dimensional turbulence where is kinematic viscosity. are mainly limited to the models with unbounded condi- vu Vorticity is defined as . Equation (1) tion, or with the identical bounded condition such as squ- x y are or circular domains. The literatures of two-dimens- can be reduced to yield a second-order equation for the ional channel flow are rare. W. Kramer, H. J. H. Clercx vorticity as follows: and G. J. F. van Heijst have done some pioneering study on this subject. They have researched the influence of the 1 f f 2 1 3 (3) aspect ratio of the channel and the integral-scale Rey- tzRe x nolds number on the large-scale self-organization of the flow in detail and obtained lots of important consequence The velocity component equations can be deduced due [11]. In the present paper, the numerical process to to and continuity Equation (2): simulate the two-dimensional turbulent channel flows directly with spectral method is firstly introduced, and 2u , 2v (4) the accuracy and stability of the proposed algorithm is y x verified with two examples. Secondly the relation be- Fully developed turbulent channel flow is homogene- tween the wall shear stress and near-wall vortices is ex- ous in the streamwise direction, and periodic boundary plored and the maximum correlation-value location of conditions are used in this direction. All unknown quan- the near-wall vortices is obtained. Finally the instanta- tities are expanded with Fourier series in the streamwise neous diagrams of fluctuation velocity vectors and the direction and Chebyshev polynomial in the normal direc- near-wall model are analyzed to provide a further ex- tion as follows: pression for the conclusions obtained. NN/2 12 qxyt(,,) qmp ( t )exp( imp xT ) ( y ) 2．Numerical Processes mN1 /2 P 0 where m . is amount of streamwise period, 2.1. Numerical Method m defined as 2/ x , x is the non-dimensional width n n With the development of computational technology and of computational domain in the streamwise direction; m resources, the direct numerical simulation (DNS) is more is streamwise wave number; Typ () is p-order Cheby- and more widely used as the basic research approach of shev polynomial. Copyright © 2010 SciRes. ENG 192 S. X. Guo ET AL. Substitute the expansions of all unknown quantities NN22 ˆˆp into Equations (3) and (4) respectively, and the spectral vtmp( ) 0, ( 1)vt mp ( ) 0 (9) coefficient equations for each Fourier wave number can pp00 be obtained: Equations of vorticity-spectral coefficient and veloci- tiey-spectral coefficients with corresponding boundary N2 1 22ˆ conditions compose to respective closed equations sets, [(mmppDtT )]()()y p0 t Re which can be solved and the corresponding spectral co- N efficients can be obtained. The time advancement is car- 2 ˆˆ ried out by semi-implicit scheme: Crank-Nicolsion for [()f1,mptD i m f2, mp ()]( t T p y) (5) p0 the viscous terms and Adams-Bashforth for the nonlinear terms. The detailed discrete process can be consulted in NN22Reference [14]. 22ˆ ˆ (mmppmppDu ) () tTy ( ) () tDTy ( ) (6) pp00 2.2. Computational Model NN22 22ˆ ˆ ()()()()mmppmmppDv tTy i tTy() (7) pp00Two-dimensional channel flow is chosen as the numeri- cal model. The flow geometry and the coordinate system d are shown in Figure 1. The non-dimensional size of where differential operator D defines as D . Equa- dy computational domain is [0,2 ] [1,1] .