Author | Starke, A.R. |

Title | A combined experimental and numerical study of the turbulent wake |

Conference/Journal | PhD-thesis, Delft University of Technology |

Month | January |

Year | 2002 |

Abstract

The main objective of the present investigation is to gain a better understanding of the flow

in the far and near wake of a flat plate. To realize this objective a study is made of three

flow configurations. First, the asymptotic far wake is studied, where, according to classical

theory, properly scaled flow quantities become independent of the strearnwise distance.

Experimental data available in the literature are used to validate numerical simulations

of the far wake using two algebraic turbulence models and two two-equation models (k - E

and k - w). Furthermore, new experimental data are obtained in the near wake of a

flat plate that is subjected to a nominally zero pressure gradient. These experimental

results are compared with numerical simulations using both a k - E turbulence model

and a differential Reynolds-stress model. Finally a similar experimental and numerical

investigation is made of the effects of curvature of the mean streamlines on the wake of

the plate.

In the numerical investigation of the far wake a study is made of the self-similar solutions

obtained with various turbulence models for a wake subjected to a zero pressure gradient.

In order to calculate these solutions, the boundary-layer equations are further simplified

using the conditions on the existence of similarity solutions. As a result, the partial

differential equations that describe the flow in a wake simplify to ordinary differential

equations, which are solved numerically. The calculations show that both investigated

two-equation models have discontinuous solutions at the interface between the wake and

the surrounding free stream. This is confirmed analytically by a power-law approximation

of the solution near this interface. The analysis also shows that the discontinuity is

directly related to the relative magnitudes of the model constants in the diffusive terms

of the turbulence equations. The numerical simulations of the far wake are furthermore

used to study the strong dependency of the results obtained with the k - w model on the

free-stream boundary condition for ta,

With respect to the experimental investigation of the far wake it is noted that based

on earlier studies by other researchers it has generally been assumed that the quantities

in the far wake are independent of the wake-generating body. Recently, however, some

critical observations have been added to this theory. The present study shows, on the

basis of various experimental results available in the literature, that both the approach

towards self-similarity and the self-similar state of a wake itself, vary between different

wake-generating bodies. This has serious consequences for numerical simulations based

on the similarity equations, since it is not possible to reproduce this experimental result

numerically with the turbulence models used in the present investigation; each of the

investigated models gives only one solution (with a given set of model constants) in the

far wake, irrespective of the wake-generating body.

In the study of the wake subjected to a nominally zero pressure gradient, a threecomponent

LDA system is used to acquire data on mean velocities and Reynolds stresses.

A clear influence is observed of the tapered trailing edge of the plate on the near wake.

iv SUMMARY

This creates an internal favourable pressure gradient in the near wake and a corresponding

decrease of the momentum-loss thickness in downstream direction. The change-over

of the wall-bounded flow on the plate into the wake causes a pronounced increase in the

level of the transverse normal-stresses in the near wake.

The experimental results are used for a detailed comparison with numerical simulations

based on the steady, incompressible Reynolds-averaged Navier-Stokes equations. In these

computations two turbulence models are used: a k - e model and a differential Reynoldsstress

model. Both models give good results for the mean velocities, the turbulent kinetic

energy and the shear stresses. The anisotropy of the normal-stress components in the near

wake is furthermore predicted fairly accurately by the Reynolds-stress model, although

the experimentally observed increase in the level of transverse normal-stresses along the

centreline in the near wake is not reproduced.

The final subject is the effect of curvature of the mean streamlines on the turbulence in

the wake. In the experimental study curvature is induced by placing one of the adjustable

side walls of the test section at a diverging angle with respect to the centreline of the wind

tunnel, while the opposite wall is placed parallel to this line. As a result a lateral pressure

gradient across the wake causes the wake to curve towards the diverging test-section

wall. The three-component LDA system is used to acquire data on mean velocities and

Reynolds stresses. The strongest curvature effects are observed close to the trailing edge

of the plate. Comparison with computations shows good agreement with the experiments

at not too large distances from the plate. However, further downstream a considerable

underestimation of the transverse shift of the wake is predicted, which is found to be very

sensitive to small variations in the applied pressure boundary conditions.

The main objective of the present investigation is to gain a better understanding of the flow

in the far and near wake of a flat plate. To realize this objective a study is made of three

flow configurations. First, the asymptotic far wake is studied, where, according to classical

theory, properly scaled flow quantities become independent of the strearnwise distance.

Experimental data available in the literature are used to validate numerical simulations

of the far wake using two algebraic turbulence models and two two-equation models (k - E

and k - w). Furthermore, new experimental data are obtained in the near wake of a

flat plate that is subjected to a nominally zero pressure gradient. These experimental

results are compared with numerical simulations using both a k - E turbulence model

and a differential Reynolds-stress model. Finally a similar experimental and numerical

investigation is made of the effects of curvature of the mean streamlines on the wake of

the plate.

In the numerical investigation of the far wake a study is made of the self-similar solutions

obtained with various turbulence models for a wake subjected to a zero pressure gradient.

In order to calculate these solutions, the boundary-layer equations are further simplified

using the conditions on the existence of similarity solutions. As a result, the partial

differential equations that describe the flow in a wake simplify to ordinary differential

equations, which are solved numerically. The calculations show that both investigated

two-equation models have discontinuous solutions at the interface between the wake and

the surrounding free stream. This is confirmed analytically by a power-law approximation

of the solution near this interface. The analysis also shows that the discontinuity is

directly related to the relative magnitudes of the model constants in the diffusive terms

of the turbulence equations. The numerical simulations of the far wake are furthermore

used to study the strong dependency of the results obtained with the k - w model on the

free-stream boundary condition for ta,

With respect to the experimental investigation of the far wake it is noted that based

on earlier studies by other researchers it has generally been assumed that the quantities

in the far wake are independent of the wake-generating body. Recently, however, some

critical observations have been added to this theory. The present study shows, on the

basis of various experimental results available in the literature, that both the approach

towards self-similarity and the self-similar state of a wake itself, vary between different

wake-generating bodies. This has serious consequences for numerical simulations based

on the similarity equations, since it is not possible to reproduce this experimental result

numerically with the turbulence models used in the present investigation; each of the

investigated models gives only one solution (with a given set of model constants) in the

far wake, irrespective of the wake-generating body.

In the study of the wake subjected to a nominally zero pressure gradient, a threecomponent

LDA system is used to acquire data on mean velocities and Reynolds stresses.

A clear influence is observed of the tapered trailing edge of the plate on the near wake.

iv SUMMARY

This creates an internal favourable pressure gradient in the near wake and a corresponding

decrease of the momentum-loss thickness in downstream direction. The change-over

of the wall-bounded flow on the plate into the wake causes a pronounced increase in the

level of the transverse normal-stresses in the near wake.

The experimental results are used for a detailed comparison with numerical simulations

based on the steady, incompressible Reynolds-averaged Navier-Stokes equations. In these

computations two turbulence models are used: a k - e model and a differential Reynoldsstress

model. Both models give good results for the mean velocities, the turbulent kinetic

energy and the shear stresses. The anisotropy of the normal-stress components in the near

wake is furthermore predicted fairly accurately by the Reynolds-stress model, although

the experimentally observed increase in the level of transverse normal-stresses along the

centreline in the near wake is not reproduced.

The final subject is the effect of curvature of the mean streamlines on the turbulence in

the wake. In the experimental study curvature is induced by placing one of the adjustable

side walls of the test section at a diverging angle with respect to the centreline of the wind

tunnel, while the opposite wall is placed parallel to this line. As a result a lateral pressure

gradient across the wake causes the wake to curve towards the diverging test-section

wall. The three-component LDA system is used to acquire data on mean velocities and

Reynolds stresses. The strongest curvature effects are observed close to the trailing edge

of the plate. Comparison with computations shows good agreement with the experiments

at not too large distances from the plate. However, further downstream a considerable

underestimation of the transverse shift of the wake is predicted, which is found to be very

sensitive to small variations in the applied pressure boundary conditions.