Experimental measurements are presented showing the effects
of streamline convergence on developing turbulent boundary layers. The
longitudinal pressure-gradient
in these experiments is nominally zero so the only extra rate-of-strain
is the lateral
convergence. Measurements have been made of mean flow and turbulence quantities
at two different Reynolds numbers. The results show that convergence leads
to a
significant reduction in the skin-friction and an increase in the boundary
layer
thickness. There are also large changes in the Reynolds stresses with reductions
occurring in the inner region and some increase in the outer flow.
This is in contrast
to the results of Saddoughi & Joubert (1991) for a diverging flow of
the same included
angle and zero pressure-gradient which show much smaller changes in the
stresses and
an approach to equilibrium. A new non-dimensional parameter,
βD, is proposed to
characterize the local effect of the convergence and it is shown how
this parameter is
related to Clauser's pressure-gradient parameter, βx.
It is suggested that this is an
equilibrium parameter for turbulent boundary layers with lateral straining.
In the
present flow case βD increases rapidly with
streamwise distance leading to a significant
departure from equilibrium. Measurement of terms in the transport equations
suggest
that streamline convergence leads to a reduction in production and generation
and
large increases in mean advection. The recovery of the flow after the removal
of
convergence has been shown to be characterized by a significant increase
in the
turbulent transport of shear-stress and turbulent kinetic energy from the
very near-wall
region to the flow further out where the stresses have been depleted by
convergence.
The problem of forced convection transpired turbulent boundary layers with external pressure gradient has been studied by using different scalings proposed by various researchers. Three major results were obtained: First, for adverse pressure gradient boundary layers with suction, the mean deficit profiles collapse with the free stream velocity, U∞, but into different curves depending on the strength of the blowing parameter and the upstream conditions. Second, the dependencies on the blowing parameter, the Reynolds number, and the strength of pressure gradient are removed from the outer flow when the mean deficit profiles are normalized by the Zagarola/Smits [Zagarola, M. V., and Smits, A. J., 1998, “Mean-Flow Scaling of Turbulent Pipe Flow,” J. Fluid Mech., 373, 33–79] scaling, U∞δ*/δ. Third, the temperature profiles collapse into a single curve using the new inner and outer scalings proposed by Wang and Castillo [Wang, X., and Castillo, L., 2003, “Asymptotic Solutions in Forced Convection Turbulent Boundary Layers,” J. Turbulence, 4(006)], which produce the true asymptotic profiles even at finite Pe´clet number.
An asymptotic analysis is developed for turbulent boundary layers in strong adverse pressure gradients. It is found that the boundary layer divides into three distinguishable regions: these are the wall layer, the wake layer and a transition layer. This structure has two key differences from the zero-pressure-gradient boundary layer: the wall layer is not exponentially thinner than the wake; and the wake has a large velocity deficit, and cannot be linearized. The mean velocity profile has a y½ behaviour in the overlap layer between the wall and transition regions.The analysis is done in the context of eddy viscosity closure modelling. It is found that k-ε-type models are suitable to the wall region, and have a power-law solution in the y½ layer. The outer-region scaling precludes the usual ε-equation. The Clauser, constant-viscosity model is used in that region. An asymptotic expansion of the mean flow and matching between the three regions is carried out in order to determine the relation between skin friction and pressure gradient. Numerical calculations are done for self-similar flow. It is found that the surface shear stress is a double-valued function of the pressure gradient in a small range of pressure gradients.
Experiments were done on sink flow turbulent boundary layers over a wide range of streamwise pressure gradients in order to investigate the effects on the mean velocity profiles. Measurements revealed the existence of non-universal logarithmic laws, in both inner and defect coordinates, even when the mean velocity descriptions departed strongly from the universal logarithmic law (with universal values of the Kármán constant and the inner law intercept). Systematic dependences of slope and intercepts for inner and outer logarithmic laws on the strength of the pressure gradient were observed. A theory based on the method of matched asymptotic expansions was developed in order to explain the experimentally observed variations of log-law constants with the non-dimensional pressure gradient parameter (Δp=(ν/ρU3τ)dp/dx). Towards this end, the system of partial differential equations governing the mean flow was reduced to inner and outer ordinary differential equations in self-preserving form, valid for sink flow conditions. Asymptotic matching of the inner and outer mean velocity expansions, extended to higher orders, clearly revealed the dependence of slope and intercepts on pressure gradient in the logarithmic laws.
A thorough self-similarity analysis is presented to investigate the properties of self-similarity for the outer layer of compressible turbulent boundary layers. The results are validated using the compressible and quasi-incompressible direct numerical simulation (DNS) data shown and discussed in the first part of this study; see Wenzel et al. (J. Fluid Mech., vol. 880, 2019, pp. 239–283). The analysis is carried out for a general set of characteristic scales, and conditions are derived which have to be fulfilled by these sets in case of self-similarity. To evaluate the main findings derived, four sets of characteristic scales are proposed and tested. These represent compressible extensions of the incompressible edge scaling, friction scaling, Zagarola–Smits scaling and a newly defined Rotta–Clauser scaling. Their scaling success is assessed by checking the collapse of flow-field profiles extracted at various streamwise positions, being normalized by the respective scales. For a good set of scales, most conditions derived in the analysis are fulfilled. As suggested by the data investigated, approximate self-similarity can be achieved for the mean-flow distributions of the velocity, mass flux and total enthalpy and the turbulent terms. Self-similarity thus can be stated to be achievable to a very high degree in the compressible regime. Revealed by the analysis and confirmed by the DNS data, this state is predicted by the compressible pressure-gradient boundary-layer growth parameter $\unicode[STIX]{x1D6EC}_{c}$, which is similar to the incompressible one found by related incompressible studies. Using appropriate adaption, $\unicode[STIX]{x1D6EC}_{c}$ values become comparable for compressible and incompressible pressure-gradient cases with similar wall-normal shear-stress distributions. The Rotta–Clauser parameter in its traditional form $\unicode[STIX]{x1D6FD}_{K}=(\unicode[STIX]{x1D6FF}_{K}^{\ast }/\bar{\unicode[STIX]{x1D70F}}_{w})(\text{d}p_{e}/\text{d}x)$ with the kinematic (incompressible) displacement thickness $\unicode[STIX]{x1D6FF}_{K}^{\ast }$ is shown to be a valid parameter of the form $\unicode[STIX]{x1D6EC}_{c}$ and hence still is a good indicator for equilibrium flow in the compressible regime at the finite Reynolds numbers considered. Furthermore, the analysis reveals that the often neglected derivative of the length scale, $\text{d}L_{0}/\text{d}x$, can be incorporated, which was found to have an important influence on the scaling success of common ‘low-Reynolds-number’ DNS data; this holds for both incompressible and compressible flow. Especially for the scaling of the $\bar{\unicode[STIX]{x1D70C}}\widetilde{u^{\prime \prime }v^{\prime \prime }}$ stress and thus also the wall shear stress $\bar{\unicode[STIX]{x1D70F}}_{w}$, the inclusion of $\text{d}L_{0}/\text{d}x$ leads to palpable improvements.
SummaryMeasurements are presented of the mean flow properties of some three-dimensional turbulent boundary layers re-developing after reattachment behind short separation bubbles yawed at 26.5° to the main stream. For these measurements, Rθ11varied from about 550 to 1450. It was found that, where the pressure gradient parameter (ν/ρu3τ1)∂p/∂s was not greater than about 0.05, the flow in the local external streamline direction conformed well with empirical laws for fully-attached two-dimensional layers with regard to the mean velocity profiles, shape parameter relationships and skin friction laws, giving support to the usual assumption that these two-dimensional relationships may be applied to the streamwise flow in three-dimensional layers, subject to the limitation on the pressure gradient parameter. The cross-flow profiles, on the other hand, were not generally fitted well by the often-used representations of Mager and Johnston. The variations of the kinetic energy dissipation coefficient and the entrainment rate were deduced for one of the layers, both quantities being found to be higher than those predicted by empirical relationships for conventionally-developing two-dimensional layers. However, the energy dissipation is in fair agreement with that in a similarly re-developing two-dimensional flow.
Self-similar compressible turbulent boundary layers with pressure gradients. Part 2. Self-similarity analysis of the outer layer