Direct Numerical Simulation of Self-Similar Turbulent Boundary Layers in Adverse Pressure Gradients

Direct numerical simulations of the Navier–Stokes equations have been carried out with the objective of studying turbulent boundary layers in adverse pressure gradients. The boundary layer flows concerned are of the equilibrium type which makes the analysis simpler and the results can be compared with earlier experiments and simulations. This type of turbulent boundary layers also permits an analysis of the equation of motion to predict separation. The linear analysis based on the assumption of asymptotically high Reynolds number gives results that are not applicable to finite Reynolds number flows. A different non-linear approach is presented to obtain a useful relation between the freestream variation and other mean flow parameters. Comparison of turbulent statistics from the zero pressure gradient case and two adverse pressure gradient cases shows the development of an outer peak in the turbulent energy in agreement with experiment. The turbulent flows have also been investigated using a differential Reynolds stress model. Profiles for velocity and turbulence quantities obtained from the direct numerical simulations were used as initial data. The initial transients in the model predictions vanished rapidly. The model predictions are compared with the direct simulations and low Reynolds number effects are investigated.     

Energy-balance equation for turbulent pulsations in the theory of the free turbulent boundary layer

Semiempirical expressions are proposed for the coefficient of turbulent viscosity and for the scale of turbulence in the equations for the free turbulent boundary layer in an incompressible fluid, these equations consisting of the equation of continuity, the equations of motion, and the equation for the average energy balance in the turbulent pulsations. The advantage of the expressions over the existing ones is that the two empirical constants in the equations have nearly the same values for circular and plane turbulent streams and also for a turbulent boundary layer at the edge of a semiinfinite homogeneous flow with a stationary fluid. The mean-energy distribution and the mean energy of the turbulent pulsations computed in this paper agree well with the experimental values.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 75–79, November–December, 1970.     

A quasi-similarity analysis of the turbulent boundary layer on a flat plate

The partial differential equation of the boundary layer on a flat plate are simplified by using the universal variables for turbulent flow. For laminar flow this gives boundary layer having a finite thickness and a friction coefficient differing by a few percent from the Blasius value. For a turbulent flow a differential equation for the velocity distribution is obtained with a parameter which varies slowly with the streamwise coordinate. The numerical value of this parameter is determined as an eigenvalue of the differential equations giving a velocity profile which evolves as the boundary layer thickens. Numerical calculations using a simple eddy viscosity model gave results in very good agreement with experiment.     

Influence of turbulent Prandtl number on heat transfer of a flat plate

In computations involving heat transfer in turbulent flow past bodies it is necessary to assume turbulent Prandtl number distribution across the boundary layer. A review and comparison of results obtained by different authors are given, e.g., in [1–5]. Unfortunately, the existing data are so contradictory that, at present, it does not appear to be possible to establish reliably a function that determines turbulent Prandtl number distribution across the boundary layer. The absence of sufficiently reliable and general results on the distribution of turbulent Prandtl number led to the result that in the majority of studies conducted in earlier years its value was assumed a constant and either close to or equal to one. The effect of turbulent Prandtl number on the intensity of heat transfer from a flat plate is numerically investigated in the present paper. The thermal, turbulent boundary layer equation is integrated for this purpose at different values of turbulent Prandtl number and results are compared with experimental data. Results from [6], where the thermal boundary layer was numerically integrated with Pr_t=1 and compared with experimental data, were used for comparison in the present paper. The same numerical integration procedure as in [6] was used here.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 81–85, July–August, 1984.     

DNS and scaling laws from new symmetry groups of ZPG turbulent boundary layer flow

The Lie group, or symmetry approach, developed by Oberlack (see e.g. Oberlack [26] and references therein) is used to derive new scaling laws for various quantities of a zero pressure gradient turbulent boundary layer flow. The approach unifies and extends the work done by Oberlack for the mean velocity of stationary parallel turbulent shear flows. From the two-point correlation (TPC) equations the knowledge of the symmetries allows us to derive a variety of invariant solutions (scaling laws) for turbulent flows, one of which is the new exponential mean velocity profile that is found in the mid-wake region of flat-plate boundary layers. Further, a third scaling group was found in the TPC equations for the one-dimensional turbulent boundary layer. This is in contrast to the Navier–Stokes and Euler equations, which have one and two scaling groups, respectively. The present focus is on the exponential law in the outer region of turbulent boundary layer corresponding new scaling laws for one- and two-point correlation functions. A direct numerical simulation (DNS) of a flat plate turbulent boundary layer with zero pressure gradient was performed at two different Reynolds numbers Re=750,2240. The Navier–Stokes equations were numerically solved using a spectral method with up to 140 million grid points. The results of the numerical simulations are compared with the new scaling laws. TPC functions are presented. The numerical simulation shows good agreement with the theoretical results, however only for a limited range of applicability. PACS 02.20.-a, 47.11.+j, 47.27.Nz, 47.27.Eq     

Calculation of axisymmetric twisted and nontwisted turbulent jets

The article gives the results of calculations of non-self-similar flows in turbulent jets. Use is made of the approximation of a boundary layer [1-3]; in the case of a high degree of twisting, when a zone of reverse flow forms in the initial section, the consideration is begun in a cross section corresponding to the end of the above zone. With a numerical solution the flow parameters are determined consecutively in cross sections located downstream from the starting cross section, where they are given by the conditions of the problem. The article gives a generalized Prandtl formula for the turbulent viscosity for the cases of the flows under consideration. The results of calculations carried out using this formula are compared with experimental data. The corresponding experimental constants are determined. An integral theory is proposed describing twisted jet flows with a weak deformation of the profiles of the gas-dynamic parameters.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 71–80, May–June, 1972.     

Universal Velocity Defect Law for the Turbulent Boundary Layer

Turbulent plane boundary layer flows of an incompressible fluid are considered. A refinement of the known Coles wake law is proposed. This refinement makes it possible to ensure the smooth matching of the turbulent boundary layer velocity profile with the outer flow and to extend the range of validity of the law to the case of large positive pressure gradients. The accuracy of the analytical approximation obtained is verified by comparison with the known experimental equilibrium velocity profiles. Using the approximation proposed, a relation for calculating the cross-sectional distribution of the Reynolds stress in the equilibrium boundary layer is derived. The pressure distributions for which the equilibrium turbulent boundary layer flows are single- and two-valued are distinguished.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, 2005, pp. 89–101.Original Russian Text Copyright © 2005 by Mikhailov.     

Periodic flow and heat transfer using unstructured meshes

A numerical scheme has been developed for computing fluid flow and heat transfer in periodically repeating geometries. Unstructured solution-adaptive meshes are used in a cell-centred finite volume formulation. The SIMPLE algorithm is used for pressure‒velocity coupling. For periodic flows the static pressure is decomposed into a periodic component and one that varies linearly in the streamwise direction. The latter is computed from the imposition of overall mass balance at the periodic boundary. A subiteration between the periodic pressure correction equation and the correction to the linear component is used. For heat transfer a formulation using the physical rather than the scaled temperature is employed. The scheme is applied to both laminar and turbulent computations of periodic flow and heat transfer in a variety of heat exchanger geometries; comparison with published computations and experimental data is found to be satisfactory. © 1997 John Wiley & Sons, Ltd.     

The turbulent boundary layer on the moving surface of a cylindrical body

A study is made of the problem of a two-dimensional turbulent boundary layer on the moving surface of a cylindrical body (a Rankine oval with a relative elongation of four) moving at constant velocity in an incompressible fluid. For the numerical simulation of the turbulent flow of the fluid, the boundary layer is divided into exterior and interior regions in accordance with a two-layer model, using different expressions for the coefficients of turbulent transfer for each region. A study was nade of the development of the boundary layer on the body at different speeds of the body surface and different Reynolds numbers. The following integral characteristics were found by numerical calculation: the work of friction as the body is displaced; the work expended on the movement of its surface; and, for a flow regime with separation, the work of the pressure force. In this case the following model of separation flow is assumed: beyond the singular point in the solution of the boundary layer equations that indicates the appearance of a region of reverse flow, the pressure and friction stress on the wall are constant and are determined by their values at the singular point.Translated from Izvestiya Akademii Nauk SSSH, Mekhanika Zhidkosti i Gaza, No. 5, pp. 61–67, September–October, 1984.Finally, the author would like to thank G. G. Chernyi and Yu. D. Shevelev for useful discussions and for their interest in this work.     

Numerical investigation of the development of flow in the initial section of a circular tube in the presence of a transitional boundary layer

The differential equation for turbulent viscosity [4] with empirical coefficients chosen from the conditions of best agreement with the experiment of Kholshchevnikova [5] is used for a numerical investigation of flow with heat transfer in the initial section of a circular tube for different values of the temperature factor, and also in the presence of roughness.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 95–101, May–June, 1982.     

Three-dimensional turbulent boundary layer on a body of complex shape

Flow and heat transfer problems associated with three-dimensional compressible gas flow past a body of complex shape at a small angle of attack are investigated on the basis of a finite-difference calculation. The results of a numerical solution of the equations of the three-dimensional turbulent boundary layer are presented. The effect of the leading parameters on three-dimensional flow development and heat transfer is analyzed. The characteristic flow regions in the boundary layer are found: lines of divergence and convergence on the surface, separation zones and flow interfaces. The location of the maximum values of the heat flux and friction on the surface is determined, the behavior of the limiting streamlines on the body is described, and the intensity of the secondary flows in the boundary layer is estimated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 25–35, September–October, 1986.     

Simulation of ordered structures in open turbulent flows

Extensive experimental material [1–4] indicates that ordered (coherent) structures play an important part in determining the nature of the flow, the generation of Reynolds stresses and turbulence energy, and the transport of heat, momentum, and passive admixtures in a turbulent flow. In the present paper, a model is constructed for describing coherent structures in which, given the profile of the mean velocity, one can determine the characteristic sizes, the propagation velocities, and also the frequency and amplitude characteristics of these ordered motions. The model is based on the analogy between the ordered formations and secondary flows in a subsidiary laminar flow whose velocity profile is the same as the turbulent profile of the mean velocity. The influence of small-scale pulsations is described by the introduction of the coefficient of turbulent viscosity. In the framework of the model, numerical calculations are made for two-dimensional turbulent flows in a mixing layer, a jet, and a wake behind a cylinder. The results of the calculations are compared with experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 45–52, July–August, 1981.     

Aerodynamics of end-wall boundary layer in a vortex chamber

The need for the inclusion of end-wall boundary layers in the study of the aerodynamics of vortex chambers has been frequently mentioned in the literature. However, owing to limited experimental data [1–3] with reliable information on the wall layers, the existing computational methods for end-wall boundary layers are not well-founded. The question of which parameters determine the formation of end-wall flow remains debatable. In some studies [4, 5], the vortex chambers are conditionally divided into short and long chambers. However, there is no unique opinion on the role of end-wall flows in vortex chambers of different lengths. It has also not been established for what geometric and flow parameters the chamber could be considered long or short. In the present study, as in [1, 5–8], solution is obtained for the end-wall boundary-layer equations using integral methods, considering the boundary layer in the radial direction in the form of a submerged wall jet. Such an approach made it possible to use the laws for the development of wall jets [9], and obtain fairly simple relations for integral parameters, skin friction, mass flow in the boundary layer, and other characteristics. Results are compared with available experimental data and computations of others authors; turbulent flow is considered; results for laminar boundary layer are given in [10].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 117–126, September–October, 1986.     

Self-similar reverse flows in the region of separation of a turbulent boundary layer

The Kármán momentum equation and very simple scaling arguments relating to the profiles of the turbulent tangential stresses are used to construct a self-similar reverse flow of an incompressible fluid that, depending on the values of the empirical constants occurring in it, can be realized either behind the separation point of a turbulent boundary layer or in front of its point of reattachment. The empirical constants are determined by means of several independent experimental studies on turbulent separating flows of liquids and gases at subsonic and supersonic velocities.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. A, pp. 173–177, July–August, 1988.     

一类平衡大气边界层边界条件研究

基于标准k-ε湍流模型,首先利用湍流粘度方程和剪切应力在整个边界层内恒定的假设,推导出一类耗散率表达式,并根据常用的湍动能入口剖面方程以及平均风速剖面方程,计算获得相应的耗散率方程;然后在输运方程中添加自定义源项,通过已经确定的平均速度方程、湍动能方程、耗散率方程计算得到相应输运方程的自定义源项表达式,并进行空风洞数值模拟,从而得到了一类满足平衡大气边界层的来流边界条件．通过将这种边界条件与由湍流平衡条件得到的边界条件进行比较,表明本方法获得的边界条件更适用．并且,本方法无需考虑修正壁面函数和修正湍流模型常数,因而计算更为简单,可为平衡大气边界层的研究提供一种新的思路．     

An invariant solution of the turbulent boundary-layer equations

The problem of the group stratification of the system of equations describing motion in the laminar sublayer and the turbulent core is considered. The fundamental group admissible by the initial system is constructed; invariant solutions constructed on one of the subgroups lead to a system of ordinary differential equations. Joining of the solutions and interchange of the equations occur at the boundary of the laminar sublayer. A class of power-law flows of a turbulent boundary layer is investigated. In the region of decelerated motion a double-valued solution is found corresponding to attached or separated flow. The commonly used integral characteristics are calculated and presented in the form of an interpolation polynomial.Translated from Zhurnal Prikladnoi Mekhaniki i Teknicheskoi Fiziki, No. 4, pp. 126–132, July–August, 1975.     

Collapse of a cylindrical cavity in a fluid layer under impact

The case of impact on a thin annular fluid layer with a gas-filled cavity is considered. The solution of the problem reduces to integrating a system of two first-order ordinary differential equations. The equations are analyzed qualitatively, and some exact solutions are found. Cases are noted of pulsation of the cavity, and the influence of counter-pressure and viscosity is investigated. The experimental results obtained are in agreement with the numerical computations carried out herein.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 98–106, November–December, 1970.The authors are grateful to A. M. Kogan and L. V. Mostovaya for performing the computations.     

Investigation by experiment and calculation of a supersonic stream adjacent to a wall in an accompanying supersonic flow

In well-known papers devoted to the investigation of supersonic streams adjacent to a wall, the authors, as a rule, restrict themselves to the case of a subsonic blast. In the present paper we determine the velocity field and the concentration field of an admixture of helium in a plane supersonic stream of air (M₁=2.18), propagating along a surface in an accompanying supersonic flow of air (M₂=2.7 and 3.8). In the boundary layer approximation a numerical calculation is made of the non-self-similar isobaric flow, using the equation for the turbulent viscosity [1] as the closing relationship. Results of the calculation are compared with experimental data.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 25–32, July–August, 1972.     

Use of the generalized similarity method for calculating the inner zone of a turbulent boundary layer

The use of the generalized similarity method for calculating laminar boundary layers has been fully justified (see [1, §113, 114, 148]). The replacement of the partial differential equations by ordinary differential equations, their universality and the possibility of physically interpreting the solutions in the first, parametric stage of the calculations, which distinguish the generalized similarity method from direct numerical integration methods, are preserved in the case of a turbulent boundary layer also. A comparison of the calculated and experimental velocity profiles in the inner zone of the turbulent boundary layer suggests that the generalized similarity method could be used for calculating the turbulent layer as a whole.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 25–34, September–October, 1990.     

Turbulent flow and convective heat transfer in a wavy wall channel

This paper reports the numerical modeling of turbulent flow and convective heat transfer over a wavy wall using a two equations eddy viscosity turbulence model. The wall boundary conditions were applied by using a new zonal modeling strategy based on DNS data and combining the standard k– turbulence model in the outer core flow with a one equation model to resolve the near-wall region.It was found that the two-layer model is successful in capturing most of the important physical features of a turbulent flow over a wavy wall with reasonable amount of memory storage and computer time. The predicted results show the shortcomings of the standard law of the wall for predicting such type of flows and consequently suggest that direct integrations to the wall must be used instead. Moreover, Comparison of the predicted results of a wavy wall with that of a straight channel, indicates that the averaged Nusselt number increases until a critical value is reached where the amplitude wave is increased. However, this heat transfer enhancement is accompanied by an increase in the pressure drop.