The paper is devoted to an adjoint complement to the universal Law of the Wall (LoW) for fluid dynamic momentum boundary layers. The latter typically follows from a strongly simplified, unidirectional shear flow under a constant stress assumption. We first derive the adjoint companion of the simplified momentum equation, while distinguishing between two strategies. Using mixing-length arguments, we demonstrate that the frozen turbulence strategy and a LoW-consistent (differentiated) approach provide virtually the same adjoint momentum equations, that differ only in a single scalar coefficient controlling the inclination in the logarithmic region. Moreover, it is seen that an adjoint LoW can be derived which resembles its primal counterpart in many aspects. The strategy is also compatible with wall-function assumptions for prominent RANS-type two-equation turbulence models, which ground on the mixing-length hypothesis. As a direct consequence of the frequently employed assumption that all primal flow properties algebraically scale with the friction velocity, it is demonstrated that a simple algebraic expression provides a consistent closure of the adjoint momentum equation in the logarithmic layer. This algebraic adjoint closure might also serve as an approximation for more general adjoint flow optimization studies using standard one- or two-equation Boussinesq-viscosity models for the primal flow. Results obtained from the suggested algebraic closure are verified against the primal/adjoint LoW formulations for both, low- and high-Re settings. Applications included in this paper refer to two- and three-dimensional shape optimizations of internal and external engineering flows. Related results indicate that the proposed adjoint algebraic turbulence closure accelerates the optimization process and provides improved optima in comparison to the frozen turbulence approach.
A sensitive porosity adjoint method (SPAM) for optimizing the topology of fluid machines has been proposed. A sensitivity function with respect to the porosity has been developed. In the first step of the optimization process, porous media are introduced into the flow regime according to the sensitivity function. Then the optimized porous media are transformed to solid walls. The turbulent flow in porous media is accounted for by a modified eddy-viscosity based turbulence model. Its influence on the adjoint equations is nevertheless neglected, which refers to the so called frozen turbulence as…
The manuscript is concerned with a continuous adjoint complement to two-dimensional, incompressible, first-order boundary-layer equations for a flat plate boundary-layer. The text is structured into three parts. The first part demonstrates, that the adjoint complement can be derived in two ways, either following a first simplify then derive or a first derive and then simplify strategy. The simplification step comprises the classical boundary-layer (b.-l.) approximation and the derivation step transfers the primal flow equation into a companion adjoint equation. The second part of the paper com…
The paper is concerned with an adjoint complement to the Volume-of-Fluid (VoF) method for immiscible two-phase flows, e.g. air and water, which is widely used in marine engineering due to its computational efficiency. The particular challenge of the primal and the corresponding adjoint VoF-approach refers to the sharp interface treatment featuring discontinuous physical properties. Both the continuous adjoint two-phase system (integration-by-parts) and the corresponding dual compressive convection schemes (summation-by-parts) are derived for two prominent compressive convection schemes, namely…
Estimation of the initial state of turbulent channel flow from limited data is investigated using an adjoint-variational approach. The data are generated from a reference direct numerical simulation (DNS) which is sub-sampled at different spatiotemporal resolutions. When the velocity data are at 1/4096 the spatiotemporal resolution of DNS, the correlation coefficient between the true and adjoint-variational estimated state exceeds 99 percent. The robustness of the algorithm to observation noise is demonstrated. In addition, the impact of the spatiotemporal density of the data on estimation qua…
This Technical Note outlines an adjoint complement to a critical building block of pressure-based Finite-Volume (FV) flow solvers that employ a collocated variable arrangement to simulate virtually incompressible fluids. The focal point is to strengthen the adjoint pressure-velocity coupling by using an adjoint Momentum-Weighted Interpolation (MWI) strategy. To this end, analogies of established primal MWI techniques are derived and investigated. The study reveals the merits of an adjoint MWI but also highlights the importance of its careful implementation.
This work shows how the early stages of perturbation growth in a viscosity-stratified flow are different from those in a constant-viscosity flow, and how nonlinearity is a crucial ingredient. We derive the viscosity-varying adjoint Navier-Stokes equations, where gradients in viscosity force both the adjoint momentum and the adjoint scalar (here temperature). By the technique of direct-adjoint looping, we obtain the nonlinear optimal perturbation which maximises the perturbation kinetic energy of the nonlinear system. While we study three-dimensional plane Poiseuille (channel) flow with the wal…
Adjoint-based sensitivity analysis is routinely used today to assess efficiently the effect of open-loop control on the linear stability properties of unstable flows. Sensitivity maps identify regions where small-amplitude control is the most effective, i.e. yields the largest first-order (linear) eigenvalue variation. In this study an adjoint method is proposed for computing a second-order (quadratic) sensitivity operator, and applied to the flow past a circular cylinder, controlled with a steady body force or a passive device model. Maps of second-order eigenvalue variations are obtained, wi…
This paper reports on the derivation and implementation of a shape optimization procedure for the minimization of hemolysis induction in biomedical devices. Hemolysis is a blood damaging phenomenon that may occur in mechanical blood-processing applications where large velocity gradients are found. An increased level of damaged blood can lead to deterioration of the immune system and quality of life. It is, thus, important to minimize flow-induced hemolysis by improving the design of next-generation biomedical machinery. Emphasis is given to the formulation of a continuous adjoint complement to…
Two-dimensional (2D) flows are efficiently controlled with spanwise waviness, i.e. spanwise-periodic (SP) wall blowing/suction/deformation. We tackle the global linear stability of 2D flows subject to small-amplitude 3D SP control. Building on previous work for parallel flows (Boujo et al. 2015), an adjoint method is proposed for computing the 2nd-order eigenvalue sensitivity. Such control has a zero 1st-order linear effect, so the 2nd-order quadratic effect prevails. The sensitivity operator allows one (i) to predict the effect of any control without computing the controlled flow, (ii) to com…
The semi-analytical wall boundary conditions present a mathematically rigorous framework to prescribe the influence of solid walls in SPH for fluid flows. In this paper they are investigated with respect to the skew-adjoint property which implies exact energy conservation. It will be shown that this property holds only in the limit of the continuous SPH approximation, whereas in the discrete SPH formulation it is only approximately true, leading to numerical noise. This noise, interpreted as form of "turbulence", is treated using an additional volume diffusion term in the continuity equation w…
The interaction between a fluid and a wall is described with a certain boundary condition for the fluid velocity at the wall. To understand how fluids behave near a rough wall, the fluid velocity at every point of the rough surface may be provided. This approach requires detailed knowledge of, and likely depends strongly on the roughness. Another approach of modeling the boundary conditions of a rough wall is to coarse grain and extract a penetration depth over which on average the fluid penetrates into the roughness. In this work we show that for a broad range of periodic roughness patterns a…
The inner-outer interaction model (Marusic, Mathis & Hutchins, Science, vol. 329, 2010, 193-196) and the attached-eddy model (Townsend, Cambridge University Press, 1976) are two fundamental models describing the multi-scale turbulence interactions and the organization of energy-containing motions in the logarithmic region of high-Reynolds number wall-bounded turbulence, respectively. In this paper, by coupling the additive description with the attached-eddy model, the generation process of streamwise wall-shear fluctuations, resulting from wall-attached eddies, is portrayed. Then, by resorting…
Physics-Based Machine Learning Closures and Wall Models for Hypersonic Transition-Continuum Boundary Layer Predictions
Department of Aerospace and Mechanical Engineering, University of Notre Dame（航空航天与机械工程系，圣约翰大学）；Department of Mechanical and Aerospace Engineering, University of California, Irvine（机械与航空航天工程系，加州大学伊维德分校）；Mathematical Institute, University of Oxford（数学研究所，牛津大学）
Wall Shear Stress Reconstruction from Concentration: Differentiable Physics and Physics-Informed Neural Networks
Department of Mechanical Engineering, University of Utah, Salt Lake City, UT, USA（机械工程系，犹他大学，盐湖城，UT，美国）；Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, UT, USA（科学计算与成像研究所，犹他大学，盐湖城，UT，美国）；Department of Mechanical Engineering, University of Wisconsin–Milwaukee, Milwaukee, WI, USA（机械工程系，威斯康星大学密尔沃基分校，密尔沃基，WI，美国）；Department of Biomedical Engineering, University of Utah, Salt Lake City, UT, USA（生物医学工程系，犹他大学，盐湖城，UT，美国）
