Statistics for Spatial Data.

The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

Statistics For Spatial Data

高等学校計算数学学報 = Numerical mathematics

Notes on numerical fluid mechanics

In this work, modeling of the near-wall region in turbulent flows is addressed A new wall-layer model is proposed with the goal to perform high-Reynolds number large-eddy simulations of wall bounded flows in the presence of a streamwise pressure gradient The model applies both in the viscous sublayer and in the inertial region, without any parameter to switch from one region to the other An analytical expression for the velocity field as a function of the distance from the wall is derived from the simplified thin-boundary equations and by using a turbulent eddy coefficient with a damping function This damping function relies on a modified van Driest formula to define the mixing-length taking into account the presence of a streamwise pressure gradient The model is first validated by a priori comparisons with direct numerical simulation data of various flows with and without streamwise pressure gradient and with eventual flow separation Large-eddy simulations are then performed using the present wall model as wall boundary condition A plane channel flow and the flow over a periodic arrangement of hills are successively considered The present model predictions are compared with those obtained using the wall models previously proposed by Spalding, Trans ASME, J Appl Mech 28, 243 (2008) and Manhart et al, Theor Comput Fluid Dyn 22, 243 (2008)  It is shown that the new wall model allows for a good prediction of the mean velocity profile both with and without streamwise pressure gradient It is shown than, conversely to the previous models, the present model is able to predict flow separation even when a very coarse grid is used

A wall-layer model for large-eddy simulations of turbulent flows with/out pressure gradient

Shape optimization of an airfoil in a BZT flow with multiple-source uncertainties

A numerical investigation of transonic and low-supersonic flows of dense gases of the Bethe–Zel'dovich–Thompson (BZT) type is presented. BZT gases exhibit, in a range of thermodynamic conditions close to the liquid/vapour coexistence curve, negative values of the fundamental derivative of gasdynamics. This can lead, in the transonic and supersonic regime, to non-classical gasdynamic behaviours, such as rarefaction shock waves, mixed shock/fan waves and shock splitting. In the present work, inviscid and viscous flows of a BZT fluid past an airfoil are investigated using accurate thermo-physical models for gases close to saturation conditions and a third-order centred numerical method. The influence of the upstream kinematic and thermodynamic conditions on the flow patterns and the airfoil aerodynamic performance is analysed, and possible advantages deriving from the use of a non-conventional working fluid are pointed out.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A0DE3A981F667A0E3C697119C9BFBD99/S0022112007005290a.pdf/div-class-title-inviscid-and-viscous-aerodynamics-of-dense-gases-div.pdf

Inviscid and viscous aerodynamics of dense gases

A semi-intrusive deterministic approach to uncertainty quantification in non-linear fluid flow problems

Introduction D ENSE gas dynamics studies the dynamic behavior of gases in the dense regime, i.e., at temperatures and pressures close to the thermodynamic critical point. In such conditions, complex gasdynamic phenomena can appear in the transonic and supersonic regimes.1 In spite of the additional complexity of the fluid response in the dense regime, the use of dense gases is not only necessary but, in some applications, advantageous. For instance, dense gas effects play a critical role in the performance of turbomachinery and heat-transfer equipment of organic Rankine cycles (ORCs).2 This motivates the interest in developing numerical tools for the analysis and design of advanced ORC turbomachinery components. In the past, several methods for so-called “real gas flows” have been derived. Such methods were in general tailored to deal with hypersonic reacting flows, for which the use of robust upwind numerical solvers was mandatory. Unfortunately, upwind schemes require characteristic decompositions, making their realization for complex multidimensional systems quite involved. On the other hand, for nonreacting flows of gases close to saturation conditions, governed by complex equations of state, and characterized by “exotic” but quite weak waves, the use of sophisticated characteristic decompositions is not essential. For these flows, it could be more convenient to use central schemes, which, in spite of higher numerical diffusivity, have the advantage of conceptional simplicity and low computational cost. In the present work, a centered numerical solver for the computation of inviscid and viscous dense gas flows is developed. A thirdorder-accurate centered method for perfect gas flows3 is extended to the computation of dense gases. The proposed scheme is systematically compared to a well-known second-order flux-difference splitting scheme4 implemented within the same code. The computations are performed using either the van der Waals or the realistic Martin–Hou5 equation of state. The scheme is then extended to the computation of viscous dense gas flows. The fluid viscosity and thermal conductivity are evaluated using thermophysical models appropriate for gases close to saturation conditions. The proposed method is validated for several inviscid and viscous flow problems involving dense gas phenomena.

Pietro Marco Congedo

Papers

A wall-layer model for large-eddy simulations of turbulent flows with/out pressure gradient

Shape optimization of an airfoil in a BZT flow with multiple-source uncertainties

Inviscid and viscous aerodynamics of dense gases

A semi-intrusive deterministic approach to uncertainty quantification in non-linear fluid flow problems

Numerical Solver for Dense Gas Flows