We propose the use of generalized boundary layer equations and governing dimensionless parameters... more We propose the use of generalized boundary layer equations and governing dimensionless parameters for mixed convection of either the forced-convection-dominated case or the free-convection-dominated case. Such equations are derived by considering the asymptotic limits of high Reynolds number or high Peclet number, and high Rayleigh number or high Boussinesq number, with small or large Prandtl and Richardson numbers as the case may be. Here, we illustrate the ideas in the forced-convection-dominated case in the small Prandtl number limit. We show that the exercise provides a more convenient format for displaying the results of computational or experimental investigations of mixed convection, leading to more general quantitative results as long as the asymptotic conditions are met, and possibly general qualitative results even when the asymptotic conditions are not strictly satisfied. In addition the contributions of normal and tangential buoyancy are also clearly separated.
Simple Flows Using a Second Order Theory of Fluids SAMUEL PAOLUCCI, University of Notre Dame -The... more Simple Flows Using a Second Order Theory of Fluids SAMUEL PAOLUCCI, University of Notre Dame -The Navier-Stokes-Fourier (NSF) equations have proved very valuable in modeling fluid flows over the last two centuries. However, there are cases in which large gradients in velocity and/or thermal fields occur where it has been shown that they do not provide accurate results. Second order equations were derived and shown to reproduce experimental results of the shock structure of gases over a large range of Mach numbers (Paolucci & Paolucci JFM, 486, 686-710 ( )). Computer experiments using the direct simulation Monte Carlo (DSMC) method have shown that at small Knudsen number the pressure and temperature profiles in the thermal stress problem as well as in the Couette and force-driven compressible plane Poiseuille flows exhibit different qualitative behavior from the profiles obtained by NSF equations. We compare the DSMC measurements with the numerical solutions of equations resulting from the second order theory. We find that the second order equations recover many of the anomalous features (e.g., non-constant pressure and non-zero parallel heat flux). Comparisons of the predictions coming from the second order theory are provided in order to critically assess its validity and usefulness.
A mass preserving finite element formulation of the Level Set method is presented. The formulatio... more A mass preserving finite element formulation of the Level Set method is presented. The formulation is based on the discontinuous representation of the level set function and involves the Runge-Kutta Discontinuous Galerkin (RKDG) finite element method. The resulting formulation has the flexibility of treating a complicated geometry by using arbitrary triangulation. The performance of the scheme is demonstrated on a number of two-dimensional re-distance and coupled advection-redistance problems. The results indicate that the RKDG finite element formulation provides accurate solutions of the Level Set problem and has great potential in fluid dynamics applications.
We apply the Streamline Upwind Petrov Galerkin (SUPG) finite element formulation of the Level Set... more We apply the Streamline Upwind Petrov Galerkin (SUPG) finite element formulation of the Level Set method to 2D redistancing and advection problems on unstructured triangulated grids. The purpose is to test the Level Set method for mass conservation properties, where the mass is understood as the amount of fluid enclosed by the interface. For the redistancing procedure we implement the idea of mass correction suggested by Sussman and Fatemi (1999) and confirm its high accuracy within the finite element formulation. However, we find that the use of the first order SUPG formulation of the Level Set method for coupled redistancing-advection problems can result in significant loss of mass caused by distortion of the interface due to numerical diffusion. This neccesitates the use of higher order upwind finite element schemes for the advection equation.
A fully second-order continuum theory of fluids is developed. The conventional balance equations ... more A fully second-order continuum theory of fluids is developed. The conventional balance equations of mass, linear momentum, energy and entropy are used. Constitutive equations are assumed to depend on density, temperature and velocity, and their derivatives up to second order. The principle of equipresence is used along with the Coleman–Noll procedure to derive restrictions on the constitutive equations by utilizing the second law. The entropy flux is not assumed to be equal to the heat flux over the temperature. We obtain explicit results for all constitutive quantities up to quadratic nonlinearity so as to satisfy the Clausius–Duhem inequality. Our results are shown to be consistent but more general than other published results.
Spatially homogeneous reactive systems are characterised by the simultaneous presence of a wide r... more Spatially homogeneous reactive systems are characterised by the simultaneous presence of a wide range of timescales. When the dynamics of such reactive systems develop very slow and very fast timescales separated by a range of active timescales, with large gaps in the fast/active and slow/active timescales, then it is possible to attain a multiscale adaptive model reduction along with the integration of the governing ordinary differential equations using the G-Scheme framework. The G-Scheme assumes that the dynamics is decomposed into active, slow, fast, and when applicable, invariant subspaces. We derive expressions that reveal the direct link between timescales and entropy production by resorting to the estimates of the contributions of the fast and slow subspaces provided by the G-Scheme. With reference to a constant pressure adiabatic batch reactor, we compute the contribution to entropy production by the four subspaces. These numerical experiments show that, as indicated by the theoretical derivation, the contribution to entropy production of the fast subspace is of the same magnitude as the error threshold chosen for the numerical integration, and that the contribution of the slow subspace is generally much smaller than that of the active subspace. We explicitly exploit this property to identify the slow and fast subspace dimensions differently from the method adopted in the G-Scheme, where the dimensions of the subspaces are defined on the basis of the asymptotic approximations of the contributions of the fast and slow subspaces. Comparison of the outcome of the analyses performed using two types of criteria underlines the substantial equivalence of the two. This property opens the door to a number of possible applications that will be explored in future work. For example, it is possible to utilise the information on entropy production associated with reactions within each subspace to define an entropy participation index that can be utilised for model reduction.
The transfer of a multiphase fluid from a high pressure vessel to one initially at lower pressure... more The transfer of a multiphase fluid from a high pressure vessel to one initially at lower pressure is investigated. The fluid iscomposed of two phases which do not undergo any change. The phases consist of an ideal gas, and solid particles (or liquid droplets) having constant density. The mixture is assumed to be stagnant and always perfectly mixed as well as at thermal equilibrium in each constant volume vessel. The fluid also remains homogeneous and at equilibrium while flowing between vessels. The transport properties of the mixture are taken to be zero. One important finding is that the expanding mixture or pseudo-fluid behaves similar to a polytropic Abel-Noble gas. Tile mixture thermodynamic properties, the end state in each vessel at pressure equilibrium, the critical parameters, and time dependent results are given for the adiabatic and isotherma! limiting cases. The results include both initially sonic and initially subsonic transfer. No mathematical restriction is placed on the particle concentration, although some limiting results are given for small particle volume fraction. The mass transferred at adiabatic pressure equilibrium can be significantly less than that when thermal equilibrium is also reached. Furthermore, the • adiabatic pressure equilibrium level may not be the same as that obtained at therrnal equilibrium, even when all initial temperatures are the same. Finally, it is shown that the transfer times can be very slow compared to those of a pure gas due to the large reduction possible in the mixture sound , speed.
We analyze ignition phenomena by resorting to the stretching rate concept formerly introduced in ... more We analyze ignition phenomena by resorting to the stretching rate concept formerly introduced in the study of dynamical systems. We construct a Tangential Stretching Rate (TSR) parameter by combining the concepts of stretching rate with the decomposition of the local tangent space in eigen-modes. The main feature of the TSR is its ability to identify unambiguously the most energetic scale at a given space location and time instant. The TSR depends only on the local composition of the mixture, its temperature and pressure. As such, it can be readily computed during the post processing of computed reactive flow fields, both for spatially homogeneous and in-homogenous systems. Because of the additive nature of the TSR, we defined a normalized participation index measuring the relative contribution of each mode to the TSR. This participation index to the TSR can be combined with the mode amplitude participation Index of a reaction to a mode -as defined in the Computational Singular Perturbation (CSP) method -to obtain a direct link between a reaction and TSR. The reactions having both a large participation index to the TSR and a large CSP mode amplitude participation index are those contributing the most to both the explosive and relaxation regimes of a reactive system. This information can be used for both diagnostics and for the simplification of kinetic mechanisms. We verified the properties of the TSR with reference to three nonlinear planar models (one for isothermal branched-chain reactions, one for a non-isothermal, one-step system, and for non-isothermal branched-chain reactions), to one planar linear model (to discuss issues associated with non-normality), and to test problems involving hydro-carbon oxidation kinetics. We demonstrated that the reciprocal of the TSR parameter is the proper characteristic chemical time scale in problems involving multi-step chemical kinetic mechanisms, because (i) it is the most relevant time scale during both the explosive and relaxation regimes, and (ii) it is intrinsic to the kinetics, that is, it can be identified without the need of any ad-hoc assumption.
Transport Phenomena in Materials Processing and Manufacturing
A general multidimensional model for a solidifying columnar eutectic alloy is presented in which ... more A general multidimensional model for a solidifying columnar eutectic alloy is presented in which a velocity-dependent freezing temperature is coupled with the macroscale energy equation. At solidification rates (∼ 1–10 mm/s) that are representative of gravity permanent mold and die casting processes, near-eutectic alloys freeze with a macroscopically discrete solid-liquid interface at a temperature below the equilibrium eutectic temperature which can impact the heat transfer process. The model is illustrated with one-dimensional solidification in a finite domain and solved numerically with a Galerkin finite element method. By nondimensionalizing the governing equations the effect of coupled eutectic growth on heat transport is clearly identified so that the model’s sensitivity to important parameters can be investigated. Additionally, the average eutectic spacing can be determined with the temperature field, rather than post-determination from a standard, uncoupled solution of the e...
By means of weakly nonlinear analysis, we investigate the interaction between two physically dist... more By means of weakly nonlinear analysis, we investigate the interaction between two physically distinct instability modes arising in the non-Boussinesq convection flow in a differentially heated tall vertical air-filled cavity. It is shown that in the neighborhood of the codimention-2 point the primary parallel flow becomes unstable due to both shear and buoyant disturbances. The flow dynamics is modeled by a system of the two coupled Landau equations. Different possible instability wave patterns are found, and the parameter regions of their existence are discussed. Energy analysis of the interacting instability modes is also presented.
(pWAMR) method is used to simulate the Richtmyer-Meshkov instability caused by a shock interactin... more (pWAMR) method is used to simulate the Richtmyer-Meshkov instability caused by a shock interacting with a density-stratified interface. The physical problem is studied in several configurations. We present results of numerical studies that investigate the influence of initial condition parameters (amplitude and wavelength of perturbations) on mixing and transition. In addition, the evaluation of turbulence statistics provides a measure of the mixing across the scales and the correlation with the initial condition parameters. The problem is modeled using the compressible reactive Navier-Stokes equations for a gas mixture, including multi-component diffusion, Soret and Dufour effects, and state dependent thermodynamic and transport properties. Since the amplitudes of wavelets provide a direct measure of the local error, the method is able to efficiently capture to any desired accuracy a wide range of spatial scales using a relatively small number of degrees of freedom by evolving the dynamically adaptive grid. In an effective fashion, the multilevel structure of the algorithm provides a simple way to adapt computational refinements to local demands of the solution, thus automatically producing verified solutions.
Submitted for the DFD16 Meeting of The American Physical Society A Second Order Continuum Theory ... more Submitted for the DFD16 Meeting of The American Physical Society A Second Order Continuum Theory of Fluids-Beyond the Navier-Stokes Equations SAMUEL PAOLUCCI, Univ of Notre Dame-The Navier-Stokes equations have proved very valuable in modeling fluid flows over the last two centuries. However, there are some cases where it has been demonstrated that they do not provide accurate results. In such cases, very large variations in velocity and/or thermal fields occur in the flows. It is recalled that the Navier-Stokes equations result from linear approximations of constitutive quantities. Using continuum mechanics principles, we derive a second order constitutive theory that application of which should provide more accurate results is such cases. One important case is the structure of gas-dynamic shock waves. It has been demonstrated experimentally that the Navier-Stokes formulation yields incorrect shock profiles even at moderate Mach numbers. Current continuum theories, and indeed most statistical mechanics theories, that have been advanced to reconcile such discrepancies have not been fully successful. Thus, application of the second order theory based solely on a continuum formulation provides an excellent test problem. Results of the second-order equations applied to the shock structure are obtained for monatomic and diatomic gases over a large range of Mach numbers and are compared to experimental results.
Chapter 3 summarized the highlights of the concepts behind the CSP method and the TSR analysis. I... more Chapter 3 summarized the highlights of the concepts behind the CSP method and the TSR analysis. In this chapter, we will discuss a few applications of these techniques.
We propose the use of generalized boundary layer equations and governing dimensionless parameters... more We propose the use of generalized boundary layer equations and governing dimensionless parameters for mixed convection of either the forced-convection-dominated case or the free-convection-dominated case. Such equations are derived by considering the asymptotic limits of high Reynolds number or high Peclet number, and high Rayleigh number or high Boussinesq number, with small or large Prandtl and Richardson numbers as the case may be. Here, we illustrate the ideas in the forced-convection-dominated case in the small Prandtl number limit. We show that the exercise provides a more convenient format for displaying the results of computational or experimental investigations of mixed convection, leading to more general quantitative results as long as the asymptotic conditions are met, and possibly general qualitative results even when the asymptotic conditions are not strictly satisfied. In addition the contributions of normal and tangential buoyancy are also clearly separated.
Simple Flows Using a Second Order Theory of Fluids SAMUEL PAOLUCCI, University of Notre Dame -The... more Simple Flows Using a Second Order Theory of Fluids SAMUEL PAOLUCCI, University of Notre Dame -The Navier-Stokes-Fourier (NSF) equations have proved very valuable in modeling fluid flows over the last two centuries. However, there are cases in which large gradients in velocity and/or thermal fields occur where it has been shown that they do not provide accurate results. Second order equations were derived and shown to reproduce experimental results of the shock structure of gases over a large range of Mach numbers (Paolucci & Paolucci JFM, 486, 686-710 ( )). Computer experiments using the direct simulation Monte Carlo (DSMC) method have shown that at small Knudsen number the pressure and temperature profiles in the thermal stress problem as well as in the Couette and force-driven compressible plane Poiseuille flows exhibit different qualitative behavior from the profiles obtained by NSF equations. We compare the DSMC measurements with the numerical solutions of equations resulting from the second order theory. We find that the second order equations recover many of the anomalous features (e.g., non-constant pressure and non-zero parallel heat flux). Comparisons of the predictions coming from the second order theory are provided in order to critically assess its validity and usefulness.
A mass preserving finite element formulation of the Level Set method is presented. The formulatio... more A mass preserving finite element formulation of the Level Set method is presented. The formulation is based on the discontinuous representation of the level set function and involves the Runge-Kutta Discontinuous Galerkin (RKDG) finite element method. The resulting formulation has the flexibility of treating a complicated geometry by using arbitrary triangulation. The performance of the scheme is demonstrated on a number of two-dimensional re-distance and coupled advection-redistance problems. The results indicate that the RKDG finite element formulation provides accurate solutions of the Level Set problem and has great potential in fluid dynamics applications.
We apply the Streamline Upwind Petrov Galerkin (SUPG) finite element formulation of the Level Set... more We apply the Streamline Upwind Petrov Galerkin (SUPG) finite element formulation of the Level Set method to 2D redistancing and advection problems on unstructured triangulated grids. The purpose is to test the Level Set method for mass conservation properties, where the mass is understood as the amount of fluid enclosed by the interface. For the redistancing procedure we implement the idea of mass correction suggested by Sussman and Fatemi (1999) and confirm its high accuracy within the finite element formulation. However, we find that the use of the first order SUPG formulation of the Level Set method for coupled redistancing-advection problems can result in significant loss of mass caused by distortion of the interface due to numerical diffusion. This neccesitates the use of higher order upwind finite element schemes for the advection equation.
A fully second-order continuum theory of fluids is developed. The conventional balance equations ... more A fully second-order continuum theory of fluids is developed. The conventional balance equations of mass, linear momentum, energy and entropy are used. Constitutive equations are assumed to depend on density, temperature and velocity, and their derivatives up to second order. The principle of equipresence is used along with the Coleman–Noll procedure to derive restrictions on the constitutive equations by utilizing the second law. The entropy flux is not assumed to be equal to the heat flux over the temperature. We obtain explicit results for all constitutive quantities up to quadratic nonlinearity so as to satisfy the Clausius–Duhem inequality. Our results are shown to be consistent but more general than other published results.
Spatially homogeneous reactive systems are characterised by the simultaneous presence of a wide r... more Spatially homogeneous reactive systems are characterised by the simultaneous presence of a wide range of timescales. When the dynamics of such reactive systems develop very slow and very fast timescales separated by a range of active timescales, with large gaps in the fast/active and slow/active timescales, then it is possible to attain a multiscale adaptive model reduction along with the integration of the governing ordinary differential equations using the G-Scheme framework. The G-Scheme assumes that the dynamics is decomposed into active, slow, fast, and when applicable, invariant subspaces. We derive expressions that reveal the direct link between timescales and entropy production by resorting to the estimates of the contributions of the fast and slow subspaces provided by the G-Scheme. With reference to a constant pressure adiabatic batch reactor, we compute the contribution to entropy production by the four subspaces. These numerical experiments show that, as indicated by the theoretical derivation, the contribution to entropy production of the fast subspace is of the same magnitude as the error threshold chosen for the numerical integration, and that the contribution of the slow subspace is generally much smaller than that of the active subspace. We explicitly exploit this property to identify the slow and fast subspace dimensions differently from the method adopted in the G-Scheme, where the dimensions of the subspaces are defined on the basis of the asymptotic approximations of the contributions of the fast and slow subspaces. Comparison of the outcome of the analyses performed using two types of criteria underlines the substantial equivalence of the two. This property opens the door to a number of possible applications that will be explored in future work. For example, it is possible to utilise the information on entropy production associated with reactions within each subspace to define an entropy participation index that can be utilised for model reduction.
The transfer of a multiphase fluid from a high pressure vessel to one initially at lower pressure... more The transfer of a multiphase fluid from a high pressure vessel to one initially at lower pressure is investigated. The fluid iscomposed of two phases which do not undergo any change. The phases consist of an ideal gas, and solid particles (or liquid droplets) having constant density. The mixture is assumed to be stagnant and always perfectly mixed as well as at thermal equilibrium in each constant volume vessel. The fluid also remains homogeneous and at equilibrium while flowing between vessels. The transport properties of the mixture are taken to be zero. One important finding is that the expanding mixture or pseudo-fluid behaves similar to a polytropic Abel-Noble gas. Tile mixture thermodynamic properties, the end state in each vessel at pressure equilibrium, the critical parameters, and time dependent results are given for the adiabatic and isotherma! limiting cases. The results include both initially sonic and initially subsonic transfer. No mathematical restriction is placed on the particle concentration, although some limiting results are given for small particle volume fraction. The mass transferred at adiabatic pressure equilibrium can be significantly less than that when thermal equilibrium is also reached. Furthermore, the • adiabatic pressure equilibrium level may not be the same as that obtained at therrnal equilibrium, even when all initial temperatures are the same. Finally, it is shown that the transfer times can be very slow compared to those of a pure gas due to the large reduction possible in the mixture sound , speed.
We analyze ignition phenomena by resorting to the stretching rate concept formerly introduced in ... more We analyze ignition phenomena by resorting to the stretching rate concept formerly introduced in the study of dynamical systems. We construct a Tangential Stretching Rate (TSR) parameter by combining the concepts of stretching rate with the decomposition of the local tangent space in eigen-modes. The main feature of the TSR is its ability to identify unambiguously the most energetic scale at a given space location and time instant. The TSR depends only on the local composition of the mixture, its temperature and pressure. As such, it can be readily computed during the post processing of computed reactive flow fields, both for spatially homogeneous and in-homogenous systems. Because of the additive nature of the TSR, we defined a normalized participation index measuring the relative contribution of each mode to the TSR. This participation index to the TSR can be combined with the mode amplitude participation Index of a reaction to a mode -as defined in the Computational Singular Perturbation (CSP) method -to obtain a direct link between a reaction and TSR. The reactions having both a large participation index to the TSR and a large CSP mode amplitude participation index are those contributing the most to both the explosive and relaxation regimes of a reactive system. This information can be used for both diagnostics and for the simplification of kinetic mechanisms. We verified the properties of the TSR with reference to three nonlinear planar models (one for isothermal branched-chain reactions, one for a non-isothermal, one-step system, and for non-isothermal branched-chain reactions), to one planar linear model (to discuss issues associated with non-normality), and to test problems involving hydro-carbon oxidation kinetics. We demonstrated that the reciprocal of the TSR parameter is the proper characteristic chemical time scale in problems involving multi-step chemical kinetic mechanisms, because (i) it is the most relevant time scale during both the explosive and relaxation regimes, and (ii) it is intrinsic to the kinetics, that is, it can be identified without the need of any ad-hoc assumption.
Transport Phenomena in Materials Processing and Manufacturing
A general multidimensional model for a solidifying columnar eutectic alloy is presented in which ... more A general multidimensional model for a solidifying columnar eutectic alloy is presented in which a velocity-dependent freezing temperature is coupled with the macroscale energy equation. At solidification rates (∼ 1–10 mm/s) that are representative of gravity permanent mold and die casting processes, near-eutectic alloys freeze with a macroscopically discrete solid-liquid interface at a temperature below the equilibrium eutectic temperature which can impact the heat transfer process. The model is illustrated with one-dimensional solidification in a finite domain and solved numerically with a Galerkin finite element method. By nondimensionalizing the governing equations the effect of coupled eutectic growth on heat transport is clearly identified so that the model’s sensitivity to important parameters can be investigated. Additionally, the average eutectic spacing can be determined with the temperature field, rather than post-determination from a standard, uncoupled solution of the e...
By means of weakly nonlinear analysis, we investigate the interaction between two physically dist... more By means of weakly nonlinear analysis, we investigate the interaction between two physically distinct instability modes arising in the non-Boussinesq convection flow in a differentially heated tall vertical air-filled cavity. It is shown that in the neighborhood of the codimention-2 point the primary parallel flow becomes unstable due to both shear and buoyant disturbances. The flow dynamics is modeled by a system of the two coupled Landau equations. Different possible instability wave patterns are found, and the parameter regions of their existence are discussed. Energy analysis of the interacting instability modes is also presented.
(pWAMR) method is used to simulate the Richtmyer-Meshkov instability caused by a shock interactin... more (pWAMR) method is used to simulate the Richtmyer-Meshkov instability caused by a shock interacting with a density-stratified interface. The physical problem is studied in several configurations. We present results of numerical studies that investigate the influence of initial condition parameters (amplitude and wavelength of perturbations) on mixing and transition. In addition, the evaluation of turbulence statistics provides a measure of the mixing across the scales and the correlation with the initial condition parameters. The problem is modeled using the compressible reactive Navier-Stokes equations for a gas mixture, including multi-component diffusion, Soret and Dufour effects, and state dependent thermodynamic and transport properties. Since the amplitudes of wavelets provide a direct measure of the local error, the method is able to efficiently capture to any desired accuracy a wide range of spatial scales using a relatively small number of degrees of freedom by evolving the dynamically adaptive grid. In an effective fashion, the multilevel structure of the algorithm provides a simple way to adapt computational refinements to local demands of the solution, thus automatically producing verified solutions.
Submitted for the DFD16 Meeting of The American Physical Society A Second Order Continuum Theory ... more Submitted for the DFD16 Meeting of The American Physical Society A Second Order Continuum Theory of Fluids-Beyond the Navier-Stokes Equations SAMUEL PAOLUCCI, Univ of Notre Dame-The Navier-Stokes equations have proved very valuable in modeling fluid flows over the last two centuries. However, there are some cases where it has been demonstrated that they do not provide accurate results. In such cases, very large variations in velocity and/or thermal fields occur in the flows. It is recalled that the Navier-Stokes equations result from linear approximations of constitutive quantities. Using continuum mechanics principles, we derive a second order constitutive theory that application of which should provide more accurate results is such cases. One important case is the structure of gas-dynamic shock waves. It has been demonstrated experimentally that the Navier-Stokes formulation yields incorrect shock profiles even at moderate Mach numbers. Current continuum theories, and indeed most statistical mechanics theories, that have been advanced to reconcile such discrepancies have not been fully successful. Thus, application of the second order theory based solely on a continuum formulation provides an excellent test problem. Results of the second-order equations applied to the shock structure are obtained for monatomic and diatomic gases over a large range of Mach numbers and are compared to experimental results.
Chapter 3 summarized the highlights of the concepts behind the CSP method and the TSR analysis. I... more Chapter 3 summarized the highlights of the concepts behind the CSP method and the TSR analysis. In this chapter, we will discuss a few applications of these techniques.
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Papers by Samuel Paolucci