Development of a bluff body wake under the combined influence of curvature and pressure gradient

The present experimental investigation deals with the behaviour of a wake generated by a square cylinder developing in a curved diffuser, a curved duct, a straight duct and a straight diffuser having a same pressure gradient as in the curved diffuser. This enables a systematic study of the effects of curvature and pressure gradient on wake development. It is seen that the curvature makes the wake asymmetric; the wake half width increases on the inner side and decreases on the outer side; the inner side being the region between the centreline and the wall closer to the centre of curvature and the outer side being the region between the centreline and the other wall. It causes a higher entrainment in the inner side as compared to the outer side. An adverse pressure gradient, on the other hand, causes a higher wake growth and velocity defect but reduces the rate of decay of the velocity defect. These are not altered significantly when the curvature and pressure gradient effects are combined. The curvature enhances the Reynolds stresses and the kinetic energy on the inner side and suppresses them on the outer side which makes their profiles asymmetric. These profiles become more and more asymmetric with increase in the streamwise distance. When the effects of curvature and adverse pressure gradient are combined, the profiles become further asymmetric.Department of Aerospace Engineering     

Prediction of wake in a curved duct

Experimental data on the development of an aerofoil wake in a curved stream are compared with calculations based on the k-ε model of turbulence with standard constants and with the model constant C_μ dependent on the local curvature. The mean velocity profile is asymmetric, the half-width of the wake is more on the inner side of the curved duct than on the outer side, and the turbulent shear stress decreases rapidly on the outer side. The standard k-ε model is able to satisfactorily reproduce this behaviour. Making C_μ dependent on the local radius improves the agreement on the inner side but slightly worsens it on the outer side.     

Wake-boundary layer interaction subject to convex and concave curvatures and adverse pressure gradient

Measurements of mean velocity and turbulent quantities have been carried out when the wake of a symmetrical airfoil interacts with the boundary layer on the (i) walls of a straight duct/diffuser and (ii) convex and concave walls of a curved duct/diffuser. The effects of adverse pressure gradient and of curvatures on the interaction are studied separately and in combination. Six cases are considered, viz. with (i) neither pressure gradient nor curvature, (ii) adverse pressure gradient and no curvature, (iii) and (iv) convex curvature with zero and adverse pressure gradients, respectively, (v) and (vi) concave curvature with zero and adverse pressure gradients, respectively. For the flows with curvature, the curvature parameter δ/R is 0.023, and for the flows with adverse pressure gradient, the Clauser pressure gradient parameter β is 0.62. The individual influences of adverse pressure gradient and convex and concave curvatures on the boundary layer are similar to those observed by earlier investigations. It is further observed that the combined effect of concave/convex curvature and the adverse pressure gradient causes higher turbulence intensities than the sum of the individual effects. The effect of curvature is to make the wake asymmetric, and in combination with adverse pressure gradient the asymmetry increases. It is observed that the adverse pressure gradient causes faster wake–boundary-layer interaction. Comparing measurements in a straight duct, a curved duct, a curved diffuser and a straight diffuser, it is seen that the convex curvature reduces the boundary layer thickness. The asymmetry in wake development compensates for this effect and the wake–boundary-layer interaction on a convex surface is almost the same as that on a straight surface. In the case of interaction with the boundary layer on a concave surface, the curvature increases the boundary layer thickness and causes enhanced turbulence intensities. However, the asymmetry in wake is such that the extent of wake is lower towards the boundary layer side. As a result, the wake–boundary-layer interaction on concave surface is almost the same as on a straight surface. The interaction is faster in the presence of adverse pressure gradient. Received: 16 June 2000 / Accepted: 17 May 2001     

Flow characteristics in the downstream region of a conical diffuser

Using the Navier-Stokes equations in conjunction with the k-? model of turbulence, the characteristics of flow in the region downstream of a conical diffuser with 5° angle of inclination are calculated. Two representative stations 1D₂ and 10D₂ after the diffuser exit are selected for comparison against experimental results. The calculations indicate an underestimation of mean velocity and turbulence kinetic energy at the first station, while satisfactory agreement is obtained for the mean velocity at the second station. The use of a modified k-? model sensitive to adverse pressure conditions improves the predictions considerably. The effect of inlet properties and Reynolds number on the flow characteristics at the above stations is studied using various inlet profiles and a range of Reynolds numbers based on the inlet diameter from 50 000 to 280 000.     

Development of wake of a rectangular cylinder in a curved stream

In the wake of a rectangular cylinder measurements of mean velocity and some turbulent stresses are carried out in a straight duct and in a curved duct. The difference in turbulent quantities in the wake of the body, in the straight duct an in the curved duct is significant especially in the downstream side of the wake. The shear stresses are more sensitive to curvature than the normal stresses.     

NUMERICAL PREDICTION OF SEMI-CONFINED JET IMPINGEMENT AND COMPARISON WITH EXPERIMENTAL DATA

The standard k–ε eddy viscosity model of turbulence in conjunction with the logarithmic law of the wall has been applied to the prediction of a fully developed turbulent axisymmetric jet impinging within a semi-confined space. A single geometry with a Reynolds number of 20,000 and a nozzle-to-plate spacing of two diameters has been considered with inlet boundary conditions based on measured profiles of velocity and turbulence. Velocity, turbulence and heat transfer data have been obtained using laser–Doppler anemometry and liquid crystal thermography respectively. In the developing wall jet, numerical results of heat transfer compare to within 20% of experiment where isotropy prevails and the trends in turbulent kinetic energy are predicted. However, stagnation point heat transfer is overpredicted by about 300%, which is attributed directly to the turbulence model and inapplicability of the wall function.     

The evolution of initially uniform shear flow through a nearly two-dimensional 90° curved duct

An experimental study has been made in a nearly two-dimensional 90° curved duct to investigate the effects of interaction between streamline curvature and mean strain on the evolution of turbulence. The initial uniform shear at the entrance to the curved duct was varied by an upstream shear generator to produce five different shear conditions; a uniform flow (UF), a positive weak shear (PW), a positive strong shear (PS), a negative weak shear (NW) and a negative strong shear (NS). The variations of surface pressure and the mean velocity profiles along the downstream direction under different initial shears are carefully measured. The responses of turbulent Reynolds stresses and triple velocity products to the curvature and the mean strain are also investigated. The evolution of turbulence under the curvature with the different shear conditions is described in terms of the turbulent kinetic energy and the various length scales vs the angular distance θ or a curvature parameters S _c which is defined by S _c = (U/R)/(dU/dy- U/R). The results show that the turbulent kinetic energy and the integral length scale are augmented when S _c < 0.054 whereas they are suppressed when S _c > 0.054. It is also observed that the micro-length scales of Taylor and Kolmogoroff are relatively insensitive to the curvature.     

Numerical investigation of fully developed turbulent fluid flow and heat transfer in a square duct

Three-dimensional fully developed turbulent fluid flow and heat transfer in a square duct are numerically investigated with the author's anisotropic low-Reynolds-number k-ε turbulence model. Special attenton has been given to the regions close to the wall and the corner, which are known to influence the characteristics of secondary flow a great deal. Hence, instead of the common wall function approach, the no-slip boundary condition at the wall is directly used. Velocity and temperature profiles are predicted for fully developed turbulent flows with constant wall temperature. The predicted variations of both local wall shear stress and local wall heat flux are shown to be in close agreement with available experimental data. The present paper also presents the budget of turbulent kinetic energy equation and the systematic evaluation for existing wall function forms. The commonly adopted wall function forms that are valid for two-dimensional flows are found to be inadequate for three-dimensional turbulent flows in a square duct.     

Turbulence measurements in a merged jet from two opposing curved wall jets

A two-dimensional flow generated by the interaction of two opposing, symmetric curved wall jets is investigated experimentally. The overall flow field can be divided into the curved wall jet region, the interaction region, and the merged jet region; thus, the results of the measurement are discussed to characterize these three distinct regions. For the curved wall jet region, the Reynolds stress distribution, the correlation coefficient, , and the ratio of normal stresses, , are presented and the effects of curvature and adverse pressure gradient on these distributions are discussed. The Reynolds stress distributions in the interaction region are analyzed in detail to illuminate the negative production of the turbulent kinetic energy. The developing jet in this region is found to accelerate owing to the very high pressure arising from the collision of the two wall jets. A counter-gradient shear flow situation is also observed in this interacting region. Measured data in the merged jet region are often compared to those of plane jets and the development of the merged jet is discussed in that respect. The spreading rate of the present merged jet is found to be much larger than that of the plane jets. To account for the larger spreading rate, the intermittency distribution is also investigated.List of symbols b position of y where U = U _c/2 - f turbulent/non-turbulent interface crossing rate - f _max maximum interface crossing rate - h slot height of the wall jet, 10 mm - L _u integral length scale - P, P _a static and atmospheric pressure, respectively - P _u ² production rate of longitudinal normal stress - P _v ² production rate of lateral normal stress - r radial distance from the cylinder surface - R radius of curvature of the cylinder, 100 mm - r _1/2 position of r where U=U _m/2 - U streamwise velocity - U _c centerline velocity of the merged jet - U _m maximum velocity of the curved wall jet - U ₀ exit velocity - \] Reynolds stresses - V lateral velocity in the merged jet - x distance along the centerline of the merged jet - y lateral distance from the centerline of the merged jet - intermittency factor     

Turbulent separated reattached flow in a two-dimensional curved-wall diffuser

A turbulent separation-reattachment flow in a two-dimensional asymmetrical curved-wall diffuser is studied by a two-dimensional laser doppler velocimeter. The turbulent boundary layer separates on the lower curved wall under strong pressure gradient and then reattaches on a parallel channel. At the inlet of the diffuser, Reynolds number based on the diffuser height is 1.2×10⁵ and the velocity is 25.2m/s. The results of experiments are presented and analyzed in new defined streamline-aligned coordinates. The experiment shows that after Transitory Detachment Reynolds shear stress is negative in the near-wall backflow region. Their characteristics are approximately the same as in simple turbulent shear layers near the maximum Reynolds shear stress. A scale is formed using the maximum Reynolds shear stresses. It is found that a Reynolds shear stress similarity exists from separation to reattachment and the Schofield-Perry velocity law exists in the forward shear flow. Both profiles are used in the experimental work that leads to the design of a new eddy-viscosity model. The length scale is taken from that developed by Schofield and Perry. The composite velocity scale is formed by the maximum Reynolds shear stress and the Schofield-Perry velocity scale as well as the edge velocity of the boundary layer. The results of these experiments are presented in this paper.     

Numerical prediction of a turbulent curved wake and comparison with experimental data

Numerical studies of the curved wake of a NACA 0012 airfoil of chord length 0.150 m are presented. The airfoil is placed in air at 10 m/s in the straight section of a duct of 0.457 m × 0.457 m cross‐section followed by a 90° bend with a mean radius‐to‐height ratio of 1.17. The trailing edge is located at one chord length upstream of the bend entry plane. The authors' own measurements are used to define the boundary conditions and for comparison with the predicted results. The numerical models are based on the time‐averaged, three‐dimensional conservation equations of fluid flow, incorporating the k–ε, RNG k–ε, realizable k–ε and the Reynolds stress turbulence models. The results show that the models are capable of predicting the effects of curvature on the wake development. However, quantitative differences between prediction and experiment exist. The results obtained using the Reynolds stress model show better agreement with the experimental data, compared with the k–ε based models, but not consistently for all parameters. There are also better predictions by the RNG k–ε and realizable k–ε models compared with the standard k–ε model. The predicted results using the RNG k–ε are closer to experimental data than the realizable k–ε. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical prediction of turbulent flow in a conical diffuser usingk-ε model

Fully developed incompressible turbulent flow in a conical diffuser having a total divergence angle of 8° and an area ratio of 4∶1 has been simulated by ak-ε turbulence model with high Reynolds number and adverse pressure gradient. The research has been done for pipe entry Reynolds numbers of 1.16×10⁵ and 2.93×10⁵. The mean flow velocity and turbulence energy are predicted successfully and the advantage of Boundary Fit Coordinates approach is discussed. Furthermore, thek-ε turbulence model is applied to a flow in a conical diffuser having a total divergence angle of 30° with a perforated screen. A simplified mathematical model, where only the pressure drop is considered, has been used for describing the effect of the perforated screen. The optimum combination of the resistance coefficient and the location of the perforated screen is predicted for high diffuser efficiency or the uniform velocity distribution.     

Finite element simulation of turbulent Couette–Poiseuille flows using a low Reynolds number k–ϵ model

Developing Couette–Poiseuille flows at Re=5000 are studied using a low Reynolds number k–ϵ two‐equation model and a finite element formulation. Mesh‐independent solutions are obtained using a standard Galerkin formulation and a Galerkin/least‐squares stabilized method. The predictions for the velocity and turbulent kinetic energy are compared with available experimental results and to the DNS data. Second moment closure's solutions are also compared with those of the k–ϵ model. The deficiency of eddy viscosity models to predict dissymmetric low Reynolds number channel flows has been demonstrated. Copyright © 1999 John Wiley & Sons, Ltd.     

Anisotropy measurements in the boundary layer over a flat plate with suction

Laser Doppler velocity measurements are carried out in a turbulent boundary layer subjected to concentrated wall suction (through a porous strip). The measurements are taken over a longitudinal distance of 9× the incoming boundary layer thickness ahead of the suction strip. The mean and rms velocity profiles are affected substantially by suction. Two-point measurements show that the streamwise and wall-normal autocorrelations of the streamwise velocity are reduced by suction. It is found that suction alters the redistribution of the turbulent kinetic energy k between its components. Relative to the no-suction case, the longitudinal Reynolds stress contributes more to k than the other two normal Reynolds stresses; in the outer region, its contribution is reduced which suggests structural changes in the boundary layer. This is observed in the anisotropy of the Reynolds stresses, which depart from the non-disturbed boundary layer. With suction, the anisotropy level in the near-wall region appears to be stronger than that of the undisturbed layer. It is argued that the mean shear induced by suction on the flow is responsible for the alteration of the anisotropy. The variation of the anisotropy of the layer will make the development of a turbulence model quite difficult for the flow behind suction. In that respect, a turbulence model will need to reproduce well the effects of suction on the boundary layer, if the model is to capture the effect of suction on the anisotropy of the Reynolds stresses.     

Effect of streamline curvature on flow field of a turbulent plane jet in cross-flow

The effects of the streamline curvature and finite edge velocity on the flow field of a turbulent plane jet in cross-flow are studied numerically by incorporating the curvature effect in the k–ε turbulence model. The improvement in the predictions by the streamline curvature model is assessed by comparing its prediction with those by the standard k–ε model. The predictions by both the models are compared with available experimental data. It has been observed that the performance of the k–ε model with streamline curvature modification is superior to the standard k–ε model.     

Computation of turbulent asymmetric wake

The development of asymmetric wake behind an aerofoil in turbulent incompressible flow has been computed using finite volume scheme for solving two-dimensional Navier-Stokes equations along with the k-ε model of turbulence. The results are compared with available experimental data. It is observed that the computed shift of the point of minimum velocity with distance is sensitive to the prescribed value of the normal component of velocity at the trailing edge of the aerofoil. Making the model constant C_u as a function of streamline curvature and changing the production term in the equation for ε, has only marginal influence on the results.     

Evolution of turbulence characteristics from straight to curved pipes

Fully developed, statistically steady turbulent flow in straight and curved pipes at moderate Reynolds numbers is studied in detail using direct numerical simulations (DNS) based on a spectral element discretisation. After the validation of data and setup against existing DNS results, a comparative study of turbulent characteristics at different bulk Reynolds numbers Re_b = 5300 and 11,700, and various curvature parameters κ = 0, 0.01, 0.1 is presented. In particular, complete Reynolds-stress budgets are reported for the first time. Instantaneous visualisations reveal partial relaminarisation along the inner surface of the curved pipe at the highest curvature, whereas developed turbulence is always maintained at the outer side. The mean flow shows asymmetry in the axial velocity profile and distinct Dean vortices as secondary motions. For strong curvature a distinct bulge appears close to the pipe centre, which has previously been observed in laminar and transitional curved pipes at lower Re_b only. On the other hand, mild curvature allows the interesting observation of a friction factor which is lower than in a straight pipe for the same flow rate.All statistical data, including mean profile, fluctuations and the Reynolds-stress budgets, is available for development and validation of turbulence models in curved geometries.     

Positivity preserving pointwise implicit schemes with application to turbulent compressible flat plate flow

A family of positivity preserving pointwise implicit schemes applicable to source term dominated problems is constructed, where the minimum order of spatial accuracy is one and the maximum is three. It is designed for achieving steady state numerical solutions and is constructed through the analysis of appropriate model problems, where the convective fluxes for the higher‐order members are prescribed by the Chakravarthy–Osher family of total variation diminishing (TVD) schemes. Multidimensionality is facilitated by operator splitting. Numerical experimentation confirms the stability, convergence, accuracy, positivity, and computational efficiency associated with the proposed schemes. These schemes are ideally suited to solving the low‐Reynolds number turbulent k–ϵ equations for which the positivity of k and ϵ and the presence of stiff source terms are critical issues. Hence, using a finite volume formulation of these schemes, the low‐Reynolds number Chien k–ϵ turbulence model is implemented for a flat plate geometry and a series of turbulent flow (steady state) computations are carried out to demonstrate the positivity, robustness, and reliability of the algorithm. The free‐stream and initial k and ϵ values are specified in a very simple manner. Algorithm convergence acceleration is achieved using Multigrid techniques. The k–ϵ model flow predictions are shown to be in agreement with empirical profiles. Copyright © 2001 John Wiley & Sons, Ltd.