Potential-based Formulations of the Navier-Stokes Equations and their Application

Based on a Clebsch-like velocity representation and a combination of classical variational principles for the special cases of ideal and Stokes flow a novel discontinuous Lagrangian is constructed; it bypasses the known problems associated with non-physical solutions and recovers the classical Navier-Stokes equations together with the balance of inner energy in the limit when an emerging characteristic frequency parameter tends to infinity. Additionally, a generalized Clebsch transformation for viscous flow is established for the first time. Next, an exact first integral of the unsteady, three-dimensional, incompressible Navier-Stokes equations is derived; following which gauge freedoms are explored leading to favourable reductions in the complexity of the equation set and number of unknowns, enabling a self-adjoint variational principle for steady viscous flow to be constructed. Concurrently, appropriate commonly occurring physical and auxiliary boundary conditions are prescribed, including establishment of a first integral for the dynamic boundary condition at a free surface. Starting from this new formulation, three classical flow problems are considered, the results obtained being in total agreement with solutions in the open literature. A new least-squares finite element method based on the first integral of the steady two-dimensional, incompressible, Navier-Stokes equations is developed, with optimal convergence rates established theoretically. The method is analysed comprehensively, thoroughly validated and shown to be competitive when compared to a corresponding, standard, primitive-variable, finite element formulation. Implementation details are provided, and the well-known problem of mass conservation addressed and resolved via selective weighting. The attractive positive definiteness of the resulting linear systems enables employment of a customized scalable algebraic multigrid method for efficient error reduction. The solution of several engineering related problems from the fields of lubrication and film flow demonstrate the flexibility and efficiency of the proposed method, including the case of unsteady flow, while revealing new physical insights of interest in their own right

A boundary layer computational model for predicting the flow and heat transfer in sudden expansions

Fully developed turbulent and laminar flows through symmetric planar and axisymmetric expansions with heat transfer were modeled using a finite-difference discretization of the boundary-layer equations. By using the boundary-layer equations to model separated flow in place of the Navier-Stokes equations, computational effort was reduced permitting turbulence modeling studies to be economically carried out. The continuity and momentum equations were solved in a coupled manner. The validity of the once-through calculation scheme utilizing the FLARE approximation was studied by using a multiple sweep procedure in which the FLARE approximation is removed after the first sweep;For laminar constant property flow, the equations were nodimensionalized so that the solution was independent of Reynolds number. Two different dependent hydrodynamic variable sets were tried: the primitive variable set (u-v), and the streamwise velocity stream function variable set (u-(psi)). The predictions of the boundary-layer equations were identical regardless of the variable set used. The predictions of the boundary-layer equations for parameters associated with the trapped eddy compared well with the predictions of the Navier-Stokes equations and experimental measurements for laminar isothermal flow when the Reynolds number was above 200 and the ratio of inlet to outlet channel diameter(width) was less than 1/3. The reattachment length and the flow field outside of the trapped eddy were well predicted for Reynolds numbers as low as twenty for laminar flow;The Boussinesq assumption was used to express the Reynolds stresses in terms of a turbulent viscosity. Near-wall algebraic turbulence models based on Prandtl\u27s-mixing-length model and the maximum Reynolds shear stress were compared. The near-wall models were used with the standard high-Reynolds-number k-(epsilon) turbulence model. A low-turbulent-Reynolds-number k-(epsilon) model was also investigated but found to be unsuitable for separated flow. The maximum-shear-stress near-wall model gave better predictions than the Prandtl-mixing-length models, especially for heat transfer. The predicted turbulent heat transfer is primarily dependent on the turbulence model used in the near-wall region. Globally iterating over the flow field had a more pronounced effect on the heat transfer solution than on the hydrodynamic solution

Towards a solution of the closure problem for convective atmospheric boundary-layer turbulence

We consider the closure problem for turbulence in the dry convective atmospheric boundary layer (CBL). Transport in the CBL is carried by small scale eddies near the surface and large plumes in the well mixed middle part up to the inversion that separates the CBL from the stably stratified air above. An analytically tractable model based on a multivariate Delta-PDF approach is developed. It is an extension of the model of Gryanik and Hartmann [1] (GH02) that additionally includes a term for background turbulence. Thus an exact solution is derived and all higher order moments (HOMs) are explained by second order moments, correlation coefficients and the skewness. The solution provides a proof of the extended universality hypothesis of GH02 which is the refinement of the Millionshchikov hypothesis (quasi- normality of FOM). This refined hypothesis states that CBL turbulence can be considered as result of a linear interpolation between the Gaussian and the very skewed turbulence regimes. Although the extended universality hypothesis was confirmed by results of field measurements, LES and DNS simulations (see e.g. [2-4]), several questions remained unexplained. These are now answered by the new model including the reasons of the universality of the functional form of the HOMs, the significant scatter of the values of the coefficients and the source of the magic of the linear interpolation. Finally, the closures 61 predicted by the model are tested against measurements and LES data. Some of the other issues of CBL turbulence, e.g. familiar kurtosis-skewness relationships and relation of area coverage parameters of plumes (so called filling factors) with HOM will be discussed also

Numerical Investigation of Droplet Vaporization and CH4/Air, CH4/O2 and CH4/LOXCounterflowing Spray Flames for Elevated Pressure

Study and optimization of the physical and chemical processes that are involved in many applications in science and engineering are worthwhile, to ensure the stability and efficiency of their performance. Examples are combustion process in direct injection engines, gas turbine combustors, and liquid rocket propulsion systems. First step in understanding a spray must naturally be the understanding of its basic constituents: i.e. single droplets. Hence, it is important to develop good numerical models that can predict and simulate the process of evaporation of a single droplet accurately. Thus, computational investigation of the evaporation of water droplets induced by an infrared laser beam is performed. In particular, a single spherical droplet is considered, which is suspended on horizontal and vertical glass fibers in air under atmospheric pressure. The droplet heating and evaporation are induced by a pulsed CO2 laser. The fuels in liquid rocket propulsion systems, methane and kerosene, are being discussed as alternative fuels to hydrogen because of their high energy content. Methane has some advantages compared to kerosene because of its cleaner burning characteristics. The present study contributes to an improved understanding of methane/air, methane/oxygen and methane/LOX (liquid oxygen) combustion compared to the hydrogen/oxygen system. A numerical investigation of laminar CH4/air and CH4/O2 flames is performed, where different mixtures of nitrogen and oxygen in the oxidizer stream are studied. Moreover, liquid oxygen spray flames with carrier gas methane against an oxygen stream are investigated in the counterflow configuration. The obtained results may be used in (spray) flamelet library or computations of flamelet generated manifolds in turbulent combustion. The mathematical model is based on the two-dimensional conservation equations, which are transformed into one-dimensional equations using a similarity transformation. The numerical simulation concerns the axi-symmetric configuration with an adaptive grid for the gas phase. Detailed models of all relevant processes are employed; in particular, a detailed chemical reaction mechanism is used, which comprises 35 species involving 294 elementary reactions. The chemical reaction scheme presented in this work was developed in [1]. The thermodynamic data for CH4 and O2 between 100 and 300 K are implemented for normal and elevated pressures for use in computations of cryogenic CH4/LOX combustion. For the CH4/air laminar flame, the present results are compared with results from literature to verify the mathematical model, chemical mechanism and the numerical scheme. The CH4/O2 flame is studied for elevated pressures up to 2 MPa. Both extinction strain rates and the scalar dissipation rates at stoichiometric conditions are evaluated for use in future turbulent flamelet computations. It is shown that oxygen dilution, pressure, and strain rate have a pronounced effect on flame structures, which becomes evident by studying the effects of liquid oxygen compared to gaseous oxygen on flame structure. The combustion of CH4/LOX with preceding evaporation of liquid oxygen under cryogenic conditions has displayed a significant effect of the liquid phase on gas temperature. Moreover, the spray flame is broadened with increase of initial droplet size

Mass Transfer

This book covers a wide variety of topics related to advancements in different stages of mass transfer modelling processes. Its purpose is to create a platform for the exchange of recent observations, experiences, and achievements. It is recommended for those in the chemical, biotechnological, pharmaceutical, and nanotechnology industries as well as for students of natural sciences, technical, environmental and employees in companies which manufacture machines for the above-mentioned industries. This work can also be a useful source for researchers and engineers dealing with mass transfer and related issues

Numerical Simulation of Convective-Radiative Heat Transfer

This book presents numerical, experimental, and analytical analysis of convective and radiative heat transfer in various engineering and natural systems, including transport phenomena in heat exchangers and furnaces, cooling of electronic heat-generating elements, and thin-film flows in various technical systems. It is well known that such heat transfer mechanisms are dominant in the systems under consideration. Therefore, in-depth study of these regimes is vital for both the growth of industry and the preservation of natural resources. The authors included in this book present insightful and provocative studies on convective and radiative heat transfer using modern analytical techniques. This book will be very useful for academics, engineers, and advanced students