Helge I. Andersson

Direct simulations of low-Reynolds-number turbulent flow in a rotating channel

Direct numerical simulations of fully developed pressure-driven turbulent flow in a rotating channel have been performed. The unsteady Navier–Stokes equations were written for flow in a constantly rotating frame of reference and solved numerically by means of a finite-difference technique on a 128 × 128 × 128 computational mesh. The Reynolds number, based on the bulk mean velocity U m and the channel half-width h , was about 2900, while the rotation number Ro = 2|Ω| h / U m varied from 0 to 0.5. Without system rotation, results of the simulation were in good agreement with the accurate reference simulation of Kim, Moin & Moser (1987) and available experimental data. The simulated flow fields subject to rotation revealed fascinating effects exerted by the Coriolis force on channel flow turbulence. With weak rotation ( Ro = 0.01) the turbulence statistics across the channel varied only slightly compared with the nonrotating case, and opposite effects were observed near the pressure and suction sides of the channel. With increasing rotation the augmentation and damping of the turbulence along the pressure and suction sides, respectively, became more significant, resulting in highly asymmetric profiles of mean velocity and turbulent Reynolds stresses. In accordance with the experimental observations of Johnston, Halleen & Lezius (1972), the mean velocity profile exhibited an appreciable region with slope 2Ω. At Ro = 0.50 the Reynolds stresses vanished in the vicinity of the stabilized side, and the nearly complete suppression of the turbulent agitation was confirmed by marker particle trackings and two-point velocity correlations. Rotational-induced Taylor-Gortler-like counter-rotating streamwise vortices have been identified, and the simulations suggest that the vortices are shifted slightly towards the pressure side with increasing rotation rates, and the number of vortex pairs therefore tend to increase with Ro .

Magnetohydrodynamic flow of a power-law fluid over a stretching sheet

Abstract Magnetohydrodynamic flow of an electrically conducting power-law fluid over a stretching sheet in the presence of a uniform transverse magnetic field is investigated by using an exact similarity transformation. The effect of magnetic field on the now characteristics is explored numerically, and it is concluded that the magnetic field tends to make the boundary layer thinner, thereby increasing the wall friction.

Diffusion of a chemically reactive species from a stretching sheet

Abstract The transfer of chemically reactive species in the laminar flow over an elastic plane surface is considered. The viscous flow is driven solely by the linearly stretched surface and the reactive species is emitted from this sheet and undergoes an isothermal and homogeneous one-stage reaction as it diffuses into the surrounding fluid. A similarity transformation is devised, which reduces the concentration conservation equation to an ordinary differential equation. An exact analytical solution due to Crane [Z. Angew. Math. Phys. 21, 645–647 (1970)] is adopted for the velocity, whereas the concentration field is obtained numerically. The computations showed that the principal effect of a destructive chemical reaction is to reduce the thickness of the concentration boundary layer and to increase the mass transfer rate from the stretching sheet to the surrounding fluid. This effect appeared to be more pronounced for a first-order reaction than for second- and third-order reactions. A nonuniqueness of the concentration distributions for generative first-order reactions was revealed by the computations.

An investigation of turbulent plane Couette flow at low Reynolds numbers

The turbulent structure in plane Couette flow at low Reynolds numbers is studied using data obtained both from numerical simulation and physical experiments. It is shown that the near-wall turbulence structure is quite similar to what has earlier been found in plane Poiseuille flow; however, there are also some large differences especially regarding Reynolds stress production. The commonly held view that the maximum in Reynolds stress close to the wall in Poiseuille and boundary layer flows is due to the turbulence-generating events must be modified as plane Couette flow does not exhibit such a maximum, although the near-wall coherent structures are quite similar. For two-dimensional mean flow, turbulence production occurs only for the streamwise fluctuations, and the present study shows the importance of the pressure—strain redistribution in connection with the near-wall coherent events.

Flow of a power-law fluid film on an unsteady stretching surface

Abstract Flow of a thin liquid film of a power-law fluid caused by the unsteady stretching of a surface is investigated by using a similarity transformation. This transformation reduces the unsteady boundary-layer equations to a non-linear ordinary differential equation governed by a nondimensional unsteadiness parameter S. The effect of S on the film thickness is explored numerically for different values of the power-law index n. A physical explanation for the findings is also provided.

Gravity-driven laminar film flow of power-law fluids along vertical walls

Abstract A theoretical analysis is presented which brings steady laminar film flow of power-law fluids within the framework of classical boundary layer theory. The upper part of the film, which consists of a developing viscous boundary layer and an external inviscid freestream, is treated separately from the viscous dominated part of the flow, thereby taking advantage of the distinguishing features of each flow region. It is demonstrated that the film boundary layer developing along a vertical wall can be described by a generalized Falkner-Skan type equation originally developed for wedge flow. An exact similarity solution for the velocity field in the film boundary layer is thus made available. Downstream of the boundary layer flow regime the fluid flow is completely dominated by the action of viscous shear, and fairly accurate solutions are obtained by the Von Karman integral method approach. A new form of the velocity profile is assumed, which reduces to the exact analytic solution for the fully-developed film. By matching the downstream integral method solution to the upstream generalized Falkner-Skan similarity solution, accurate estimates for the hydrodynamic entrance length are obtained. It is also shown that the flow development in the upstream region predicted by the approximate integral method closely corresponds to the exact similarity solution for that flow regime. An analytical solution of the resulting integral equation for the Newtonian case is compared with previously published results.

Numerical Study of Turbulent Plane Couette Flow at Low Reynolds Number

A direct numerical simulation of fully developed turbulent plane Couette flow has been performed. Unsteady large-scale structures, which contributed to the instantaneous energy level, were observed. These evolving and drifting vortical structures vanished after time-averaging, and the resulting mean velocity and streamwise turbulence intensity compared favourably with recent laboratory data.

Numerical solution of the laminar boundary layer equations for power-law fluids

Abstract The power-law fluid boundary layer problem is formulated analogous to a turbulent flow problem. The deviation from Newtonian rheology is accounted for by an effective viscosity replacing the so called turbulent eddy viscosity. A numerical solution procedure for turbulent boundary layer problem is adapted to laminar boundary layer flows of power-law fluids. The transformed momentum equation is written as a set of first order equations, which is solved by the implicit Keller Box finite-difference scheme for parabolic problems. This method has second-order accuracy in all variables and permits arbitrary net spacings. Newtons method is used to solve the resulting system of nonlinear difference equations. Special attention is devoted to the extra nonlinearity introduced via the effective viscosity. The application of an incomplete linearization scheme, typical for turbulent flow computations, is compared with an alternative full linearization. Numerical results are presented for the similarity boundary layer flow along a flat plate, and for the non-similar boundary layer along a cylinder surface. The results are compared with other available numerical and series expansion solutions. The rate of convergence for the usual incomplete application of Newtons method is compared with the convergence of the fully linearized scheme. It was observed that the latter led to substantial savings in computational effort, except for some similarity boundary layers of pseudoplastic fluids.

Hydrodynamic entrance length of non-newtonian liquid films

Abstract Accelerating laminar thin-film flow of power-law fluids along vertical walls is considered. The momentum integral method approach is used to predict the flow development in the hydrodynamic entrance region, and two different assumptions are employed for the streamwise velocity profile. It is demonstrated that dilatant films develop more rapidly than films of pseudoplastic fluids. The entrance length predictions compare favourably with experimental data for Newtonian films.

Turbulence Statistics of Rotating Channel Flow

Turbulence statistics of rotating channel flow are considered, including budget data for the individual non-zero components of the Reynolds stress tensor. The statistics have been comxad piled from direct numerical simulations of Kristoffersen & Andersson (1993) at six different rotation numbers ROm = 2h I n I jUm in the range from 0.01 to 0.50. Complete Reynolds stress budgets are provided for the particular cases Rom = 0.15 and ROm = 0.50, while the variation with Rom of some important flow characteristics demonstrates the significant alterations in the flow field induced by the Coriolis force.