DNS of Wall-Bounded Turbulent Flows

This book covers the topic of direct numerical simulation (DNS) of wall-bounded turbulent flow by first principle approach. It is mandatory to track the flow from laminar state to fully developed turbulent state, and which has been solved very recently for specific cases. While one can attempt DNS of Navier-Stokes equation for fully developed turbulent flow, there would be ambiguity about specifying initial and boundary conditions. Moreover, such an exercise even if it is successful, then also it may not answer most of the fundamental questions related to the evolution of the flow field. First, by erroneous numerical methods and artificial excitations, one may achieve fully developed turbulence, but that approach should be avoided. Such simulations would not enlighten one in following the physical processes during transition. A detailed review and critique of some of the methods used for DNS and large eddy simulation (LES) of transitional and turbulent flows is given in Sengupta, T. K. (In the IUTAM Symposium Proceedings on Advances in Computation, Modeling and Control of Transitional and Turbulent Flows, 2015, [24]). This is of fundamental relevance for the contents of this book. Here, we present a more fundamental approach that studies the topic by considering the flow evolution from laminar to fully developed state by a thoroughly analyzed numerical method. The aim is to computationally reproduce the physical phenomena as recorded in classical transition experiments by Schubauer and Skramstad (J Aerosp Sci 14(2):69–78, 1947, [19]) for 2D transition and Klebanoff et al. (J Fluid Mech, 12:1–34, 1962, [10]) for the 3D routes. These experiments were designed after lots of soul searching in the fluid dynamic community, when experimentalists failed to detect some wave-solutions predicted by instability studies by Heisenberg, Tollmien and Schlichting. The reader is directed to the introductory discussion in Sengupta (Instabilities of flows and transition to turbulence. CRC Press, Taylor & Francis Group, Florida, 2012, [23]) about the historic development of the subject of instability studies and a brief account will be provided later.

The governing equations of fluid mechanics are all based on identical fundamental classical principles of dynamics, namely conservation of mass, momentum and energy. Based on these principles, Claude-Louis Navier and George Gabriel Stokes, derived the governing equation for viscous fluids by applying Newton’s second law to fluid motion, along with the assumption that stresses arising in the fluids are due to diffusing viscous effects and the pressure gradient. These governing equations are known as Navier–Stokes Equation (NSE). It is to be noted that there is no fixed version of NSE, an appropriate version and the formulation has to be chosen based on the need and the characteristics of the fluid-dynamical problem to be considered. For example, for low-speed applications without heat-transfer, the appropriate version is the incompressible NSE, without considering the energy equation. For such equations, it can be shown that the kinetic energy of the fluid in a control volume is automatically conserved from the momentum equation. For flows dominated by heat-transfer, an additional energy equation is to considered which essentially determines the temperature at each point. Temperature gradient induces gradient in fluid-density, which in turn affects the flow if gravitational effects are also present. In incompressible NSE, such buoyancy effects due to temperature induced density-gradient are generally modeled via Boussinesq approximation (Sengupta TK, Bhaumik S, Bose R (2013). Phys Fluids 25: 094102, [87]). In contrast, when the flow speed is on the higher side and the compressibility effects are quite dominant, the appropriate governing equations are the compressible verisons of NSE, where all the conservation constraints (mass, momentum and energy) are to be considered.

In this chapter, linear stability and receptivity analysis of the zero-pressure gradient (ZPG) boundary layer, under the parallel flow assumption, is discussed. This assumption implies that the equilibrium flow quantities do not grow in the streamwise direction and requires solving the Orr-Sommerfeld equation (OSE) to study evolution of disturbance field in a linearized analysis. The concept of the spatio-temporal wave-front (STWF) originates from the receptivity analysis with the OSE solved for the response field. First, the simplified description of equilibrium flow in terms of a similarity solution for ZPG boundary layer is presented. Following which the OSE is derived for boundary layers, making use of the parallel flow approximation (Drazin and Reid, Hydrodynamic stability, Cambridge University Press, UK, 1981, [19], Sengupta, Instabilities of flows and transition to turbulence, CRC Press, Taylor & Francis Group, Florida, USA, 2012, [53]). This equation have been solved for the ZPG boundary layer using analytical approaches in Heisenberg (Annalen der Physik Leipzig, 379:577–627, 1924, [28]), Schlichting (Nach Gesell d Wiss z G\(\ddot{\mathrm{o}}\)tt., MPK 42:181–208, 1933, [48]), Tollmien (NACA TM 609, 1931, [71]). We instead introduce the compound matrix method, a robust method for stiff differential equation useful for the OSE. Finally, the receptivity analysis of the ZPG boundary layer flow is provided, with results taken from Sengupta et al. (Phys Rev Lett, 96(22):224504, 2006, [61]), Sengupta et al. (Phys Fluids, 18:094101, 2006, [62]). The unique feature of the materials in this chapter is the topic of instability of mixed convection flows for which two theorems are enunciated for an inviscid linear mechanism, based on materials extensively taken from Sengupta et al., Physics of Fluids, 25, 094102 (2013).

Morkovin (Transition to turbulence, ASME FED Publication, USA, vol 114, pp 1–12, 1991, [33]) classified transition to turbulence in to two main types: (i) The classical primary instability route whose onset is marked along with the presence of TS waves (as in ZPGBL) and (ii) the bypass routes, which encompass all other possible transition scenarios that do not exhibit TS waves. Unfortunately, this is too simplistic a classification scheme for the reasons given in the introduction. Moreover, the central theme of this chapter, is to show some typical bypass transition events shown experimentally and the corresponding theoretical explanations of these events. Of special interest is the development of two nonlinear theories of receptivity, derived from Navier–Stokes equation, without making any assumptions.

A general consensus among fluid dynamicists is that flow transition from laminar to turbulent state occurs due to its instability by imposed and/or background omnipresent disturbances (Drazin and Reid, Hydrodynamic stability, Cambridge University Press, UK, 1981, [7]; Sengupta, Instabilities of flows and transition to turbulence, CRC Press, USA, 2012, [17]). These ideas have prompted researchers to study the problem of flow transition from the perspective of the stability or receptivity of equilibrium flows. For a ZPGBL, first attempts include analyses by linearized inviscid and viscous instability theories. For a parallel boundary layer the latter approach gives rise to the OSE (Drazin and Reid, Hydrodynamic stability, Cambridge University Press, UK, 1981, [7]; Sengupta, Instabilities of flows and transition to turbulence, CRC Press, USA, 2012, [17]). As discussed in the previous chapters, the solution of the OSE predicts the existence of spatially modulated wavy solutions, known as TS waves (Drazin and Reid, Hydrodynamic stability, Cambridge University Press, UK, 1981, [7]). First experimental detection of spatially evolving TS wave-packets were reported by Schubauer and Skramstad (J Aeronaut Sci, 14(2), 69–78, [16]), who essentially perturbed the ZPGBL boundary layer by vibrating a ribbon at a fixed frequency inside it. Following mathematical physics, disturbance evolution in any continuum medium can occurs following temporal or spatial or spatio-temporal routes. For example, there has been an effort (Brillouin, Wave propagation and group velocity, Academic Press, New York, 1960, [4]), where wave propagation problem in electromagnetic medium has been considered with the proviso that the wave-train is preceded by a spatio-temporal wave-packet or STWF. The success of the experiments by Schubauer and Skramstad (J Aeronaut Sci, 14(2), 69–78, [16]) in detecting TS waves, prompted researchers to predominantly consider the possibility that flow transition in fluid flows (specifically for wall-bounded flows) follow spatial route following the growth of TS waves only. Thus, in fluid mechanics no effects have been made to find STWF for a long time.

In Chap. 5, we have discussed the dynamics of the STWF for 2D transition. We have also shown the inadequacy of the linear spatial instability studies in determining the evolution of disturbances. For monochromatic wall-excitation, the spatio-temporal evolution of disturbance was noted to depend on various factors like (a) excitation frequency, (b) amplitude, (c) exciter location and its width and (d) nature of excitation onset. In the present chapter, we would discuss about the 3D evolution of disturbances and the associated process of transition to turbulence. We first start with the governing equations, followed by numerical methods, problem definition and a brief description of boundary conditions. We have chosen the velocity-vorticity formulation of the incompressible NSE for its inherent accuracy to compute the 3D excitation of a nominally 2D ZPG boundary layer. Growth and evolution of disturbances, nature of vortical structures in the transitional and turbulent zones, and integral properties of the turbulent boundary layer (in terms of displacement and momentum thickness, shape factor and skin friction coefficient) are described subsequently.