Temporal slow-growth formulation for direct numerical simulation of compressible wall-bounded flows

Abstract

A new slow growth formulation for DNS of wall-bounded turbulent flow is developed and demonstrated to enable extension of slow growth modeling concepts to complex boundary layer flows.As in previous slow growth approaches, the formulation assumes scale separation between the fast scales of turbulence and the slow evolution of statistics such as the mean flow. This separation enables the development of approaches where the fast scales of turbulence are directly simulated while the forcing provided by the slow evolution is modeled. The resulting model admits periodic boundary conditions in the streamwise direction, which avoids the need for extremely long domains and complex inflow conditions that typically accompany spatially developing simulations. Further, it enables the use of efficient Fourier numerics. Unlike previous approaches [2,18], the present approach is based on a temporally evolving boundary layer and is specifically tailored to give results for calibration and validation of RANS turbulence models. The use of a temporal homogenization simplifies the modeling, enabling straightforward extension to flows with complicating features, including cold and blowing walls. To generate data useful for calibration and validation of RANS models, special care is taken to ensure that the mean slow growth forcing is closed in terms of the mean and other quantities that appear in standard RANS models, ensuring that there is no confounding between typical RANS closures and additional closures required for the slow growth problem. The performance of the method is demonstrated on two problems: an essentially incompressible, zero-pressure-gradient boundary layer and a transonic boundary layer over a cooled wall with wall transpiration. The results show that the approach produces flows that are qualitatively similar to other slow growth methods as well as spatially developing simulations and that the new method can be a useful tool in investigating complex wall-bounded flows. †

Near-wall patch representation of wall-bounded turbulence

“…A higher-order approximation could allow one or more of these quantities to vary slowly in the streamwise direction or time, which can be treated asymptotically as in Spalart (1988) or Topalian et al. (2017) to model the resulting spatial or temporal evolution of the wall turbulence. This would broaden the applicability of NWP models, which, for example, would allow them to be used with stronger pressure gradients, especially strong adverse-pressure-gradient boundary layers in which wall shear stress generally evolves relatively rapidly.…”

“…Finally, the NWP formulation described here can be considered a lowest-order asymptotic description of near-wall turbulence, in which the imposed pressure gradient, the momentum flux from the outer flow and the mean wall shear stress are considered uniform in space and time on the scale of the patch. A higher-order approximation could allow one or more of these quantities to vary slowly in the streamwise direction or time, which can be treated asymptotically as in Spalart (1988) or Topalian et al (2017) to model the resulting spatial or temporal evolution of the wall turbulence. This would broaden the applicability of NWP models, which, for example, would allow them to be used with stronger pressure gradients, especially strong adverse-pressure-gradient boundary layers in which wall shear stress generally evolves relatively rapidly.…”

Slow-growth approximation for near-wall patch representation of wall-bounded turbulence

“…The domain is periodic in the streamwise and spanwise directions, similar to the minimal flow unit simulations first conducted by Jiménez & Moin (1991). Within the domain, the pressure gradient is assumed to be constant, while streamwise growth of the mean wall-shear stress that occurs in a turbulent boundary layer, for example, is accounted for asymptotically, similar to the related works by Spalart (1988) and Topalian et al (2017). Far-field boundary conditions in the wall-normal direction are formulated by using a fringe-region (Colonius 2004) in which the mean flux of streamwise momentum in the wall-normal direction is prescribed as a function of the pressure gradient and the asymptotic growth parameter.…”

“…Another is to characterize the extent to which the dynamics of the small-scale motions, isolated from modulations by large-scale structures, are responsible for observed characteristics of near-wall turbulence. As in Spalart (1988) and Topalian et al (2017), the model equations of motion are informed by asymptotic analysis of the Navier-Stokes (NS) equations; the fundamental assumption in the current setting is a separation between the viscous length scale and the length scale over which the mean wall shear stress evolves.…”

“…To account for the relatively rapid downstream development of mean quantities in adverse-pressure-gradient boundary layers (Kitsios et al 2017), we develop in the present work a higher-order approximation that allows the mean wall shear stress to develop slowly in the streamwise direction. Similar to Spalart (1988), Guarini et al (2000), Maeder, Adams & Kleiser (2001) and Topalian et al (2017), asymptotic analysis is used to derive a set of 'homogenized' equations that describe the mean effect of streamwise development on the near-wall dynamics.…”