Analysis of unsteady flows is a very challenging topic of research. A decade ago, potential flow equations were used to predict unsteady pressures on oscillating bodies. Recognising the fact that nonlinear aerodynamics is essential to analyse unsteady flows accurately, particularly in transonic and supersonic flows, different Euler formulations operating on moving grids have emerged recently as important CFD tools for unsteady aerodynamics. Numerical solution of Euler equations on moving grids based on upwind schemes such as the ones due to van Leer and Roe have been developed for the purpose of numerical simulation of unsteady transonic and supersonic flows. In the present work, Euler computations based on yet another recent robust upwind scheme (for steady flows) namely Kinetic Flux Vector Splitting (KFVS) scheme due to Deshpande and Mandal is chosen for further development of a time accurate Euler solver to operate on problems involving moving boundaries. The development of an Euler code based on this scheme is likely to be highly useful to analyse problems of unsteady aerodynamics and computational aeroelasiticity especially when it is noted that KFVS has been found to be an extremely robust scheme for computation of subsonic, transonic, supersonic and hypersonic flows. The KFVS scheme, basically exploits the connection between the linear scalar Boltzmann equation of kinetic theory of gases and the nonlinear vector conservation law, that is, Euler equations of fluid dynamics through moment method strategy. The KFVS scheme has inherent simplicity in splitting the flux even on moving grids due to underlying particle model. The inherent simplicity of KFVS for moving grid problems is due to its relationship with the Boltzmann equation. If a surface is moving with velocity w and a particle has velocity v, then it is quite reasonable to do the splitting based on (v-w)<0 or >0. Only particles having velocity v greater than w will cross the moving surface from left to right and similar arguments hold good for particles moving in opposite direction. It is therefore quite natural to extend KFVS by splitting the Maxwellian velocity distribution at Boltzmann level based on the sign of the normal component of the relative velocity. The relative velocity is the difference between the molecular velocity (v) and the velocity of the moving surface(w). This inherent simplicity of the Kinetic Flux Vector Splitting scheme on Moving Grids (KFMG) method has prompted us to extend the same ideas to 2-D and 3-D problems leading to the present KFMG method. If w is set to zero then KFMG formulation reduces to the one corresponding to KFVS. Thus KFMG formulations axe generalisation of the KFVS formulation. In 2-D and 3-D cases, in addition to the KFMG formulation, the method to move the grids, the appropriate boundary conditions for treating moving surfaces and techniques to improve accuracy in space and time are required to be developed. The 2-D and 3-D formulations based on Kinetic Flux Vector Splitting scheme on Moving Grids method have been developed for computing unsteady flows. Between two successive time steps, the body changes its orientation in case of an oscillation or it deforms when subjected to, aerodynamic loads. In either of these cases the grid corresponding to the first time step has to be moved or regenerated around the displaced or deformed body. There are several approaches available to generate grids around moving bodies. In the present work, the 'spring analogy method' is followed to obtain grid around deflected geometries within the frame work of structured grid. Using this method, the grids are moved from previous time to the current time. This method is capable of tackling any kind of aeroelastic deformation of the body. For oscillating bodies, a suitable boundary condition enforcing the flow tangency on the body needs to be developed. As a first attempt, the body surface has been treated as an 1-D piston undergoing compression and expansion. Then, a more general Kinetic Moving Boundary Condition(KMBC) has been developed. The KMBC uses specular reflection model of kinetic theory of gases. In order to treat fixed outer boundary, Kinetic Outer Boundary Condition(KOBC) has been applied. The KOBC is more general in the sense that, it can treat different type of boundaries (subsonic, supersonic, inflow or out flow boundary). A 2-D cell-centered finite volume KFMG Euler code to operate on structured grid has been developed. The time accuracy is achieved by incorporating a fourth order Runge-Kutta time marching method. The space accuracy has been enhanced by using high resolution scheme as well as second order scheme using the method of reconstruction of fluxes. First, the KFMG Euler code has been applied to standard test cases for computing steady flows around NACA 0012 and NACA 64AQ06 airfoils in transonic flow. For these two airfoils both computational and experimental results are available in literature. It is thus possible to verify (that is, prove the claim that code is indeed solving the partial differential equations + boundary conditions posed to the code) and validate(that is, comparison with experimental results) the 2-D KFMG Euler code. Having verified and validated the 2-D KFMG Euler code for the standard test cases, the code is then applied to predict unsteady flows around sinusoidally oscillating NACA 0012 and NACA 64A006 airfoils in transonic flow. The computational and experimental unsteady results are available in literature for these airfoils for verification and validation of the present results. The unsteady lift and normal force coefficients have been predicted fairly accurately by all the CFD codes. However there is some difficulty about accurate prediction of unsteady pitching moment coefficient. Even Navier-Stokes code could not predict pitching moment accurately. This issue needs further in depth study and probably intensive computation which have not been undertaken in the present study. Next, a two degrees of £reedom(2-DOF) structural dynamics model of an airfoil undergoing pitch and plunge motions has been coupled with the 2-D KFMG Euler code for numerical simulation of aeroelastic problems. This aeroelastic analysis code is applied to NACA 64A006 airfoil undergoing pitch and plunge motions in transonic flow to obtain aeroelastic response characteristics for a set of structural parameters. For this test case also computed results are available in literature for verification. The response characteristics obtained have showed three modes namely stable, neutrally stable and unstable modes of oscillations. It is interesting to compare the value of airfoil-to-air mass ratio (Formula) obtained by us for neutrally stable condition with similar values obtained by others and some differences between them are worth mentioning here. The values of \i for neutral stability are different for different authors. The differences in values of (Formula) predicted by various authors are primarily due to differences which can be due to grid as well as mathematical model used. For example, the Euler calculations, TSP calculations and full potential calculations always show differences in shock location for the same flow problem. Changes in shock location will cause change in pressure distribution on airfoil which in turn will cause changes in values of \L for conditions of neutral stability. The flutter speed parameter(U*) has also been plotted with free stream Mach number for two different values of airfoil - to - air mass ratio. These curves shown a dip when the free stream Mach number is close to 0.855. This is referred as "Transonic Dip Phenomenon". The shock waves play a dominant role in the mechanism of transonic dip phenomenon. Lastly, cell-centered finite volume KFMG 3-D Euler code has been developed to operate on structured grids. The time accuracy is achieved by incorporating a fourth order Runge-Kutta method. The space accuracy has been enhanced by using high resolution scheme. This code has 3-D grid movement module which is based on spring analogy method. The KMBC to treat oscillating 3-D configuration and KOBC for treating 3-D outer boundary have also been formulated and implemented in the code. The 3-D KFMG Euler code has been first verified and validated for 3-D steady flows around standard shapes such as, transonic flow past a hemisphere cylinder and ONERA M6 wing. This code has also been used for predicting hypersonic flow past blunt cone-eylinder-flare configuration for which experimental data are available. Also, for this case, the results are compared with a similar Euler code. Then the KFMG Euler code has been used for predicting steady flow around ogive-cylinder-ogive configuration with elliptical cross section. The aerodynamic coefficients obtained have been compared with those of another Euler code. Thus, the 3-D KFMG Euler code has been verified and validated extensively for steady flow problems. Finally, the 3-D KFMG based Euler code has been applied to an oscillating ogive-cylinder-ogive configuration in transonic flow. This test case has been chosen as it resembles the core body of a flight vehicle configuration of interest to DRDO,India. For this test case, the unsteady lift coefficients are available in literature for verifying the present results. Two grid sizes are used to perform the unsteady calculations using the present KFMG 3-D Euler code. The hysteresis loops of lift and moment coefficients confirmed the unsteady behaviour during the oscillation of the configuration. This has proved that, the 3-D formulations are capable of predicting the unsteady flows satisfactorily. The unsteady results obtained for a grid with size of 45x41x51 which is very close to the grid size chosen in the reference(Nixon et al.) are considered for comparison. It has been mentioned in the reference that, a phase lag of (Formula) was observed in lift coefficients with respect to motion of the configuration for a free stream Mach number of 0.3 with other conditions remaining the same. The unsteady lift coefficients obtained using KFMG code as well as those available in literature are plotted for the same flow conditions. Approximately the same phase lag of (Formula) is present (for (Formula)) between the lift coefficient curves of KFMG and due to Nixon et al. The phase lag corrected plot of lift coefficient obtained by Nixon et al. is compared with the lift coefficient versus time obtained by 3-D KFMG Euler code. The two results compare well except that the peaks are over predicted by KFMG code. It is nut clear at this stage whether our results should at all match with those due to Nixon et al. Further in depth study is obviously required to settle the issue. Thus the Kinetic Flux Vector Splitting on Moving Grids has been found to be a very good and a sound method for splitting fluxes and is a generalisation of earlier KFVS on fixed grids. It has been found to be very successful in numerical simulation of unsteady aerodynamics and computational aeroelasticity.

In the quest for a high fidelity numerical scheme for CFD it is necessary to satisfy demands on accuracy, conservation, positivity and upwinding. Recently the requirement of rotational invariance has been added to this list. In the present work we are mainly interested in upwinding and rotational invariance of Least Squares Kinetic Upwind Method (LSKUM). The standard LSKUM achieves upwinding by stencil division along co-ordinate axes which is referred to as co-ordinate splitting method. This leads to symmetry breaking and rotational invariance is lost. Thus the numerical solution becomes co-ordinate frame dependent. To overcome this undesirable feature of existing numerical schemes, a new algorithm called KUMARI (Kinetic Upwind Method Avec Rotational Invariance, 'Avec' in French means 'with') has been developed. The interesting mathematical relation between directional derivative, Fourier series and divergence operator has been used effectively to achieve upwinding as well as rotational invariance and hence making the scheme truly or genuinely multidimensional upwind scheme. The KUMARI has been applied to the test case of standard 2D shock reflection problem, flow past airfoils, then to 2D blast wave problem and lastly to 2D Riemann problem (Lax's 3rd test case). The results show that either KUMARI is comparable to or in some cases better than the usual LSKUM.

The kinetic schemes, also known as Boltzmann schemes are based on the moment-method-strategy, where an upwind scheme is first developed at the Boltzmann level and after taking suitable moments we arrive at an upwind scheme for the governing Euler or Navier-Stokes equations. The Kinetic Flux Vector Splitting (KFVS)scheme, which belongs to the family of kinetic schemes is being extensively used to compute inviscid as well as viscous flows around many complex configurations of practical interest over the past two decades. To resolve many flow features accurately, like suction peak, minimising the loss in stagnation pressure, shocks, slipstreams, triple points, vortex sheets, shock-shock interaction, mixing layers, flow separation in viscous flows require an accurate and low dissipative numerical scheme. The first order KFVS method even though is very robust suffers from the problem of having much more numerical diffusion than required, resulting in very badly smearing of the above features. However, numerical dissipation can be reduced considerably by using higher order kinetic schemes. But they require more points in the stencil and hence consume more computational time and memory. The second order schemes require flux or slope limiters in the neighbourhood of discontinuities to avoid spurious and physically meaningless wiggles or oscillations in pressure, temperature or density. The limiters generally restrict the residue fall in second order schemes while in first order schemes residue falls up to machine zero. Further, pressure and density contours or streamlines are much smoother for first order accurate schemes than second order accurate schemes. A question naturally arises about the possibility of constructing first order upwind schemes which retain almost all advantages mentioned above while at the same time crisply capture the flow features. In the present work, an attempt has been made to address the above issues by developing yet another kinetic scheme, known as the low dissipative modified KFVS (m-KFVS) method based on modified CIR (MCIR) splitting with molecular velocity dependent dissipation control function. Different choices for the dissipation control function are presented. A detailed mathematical analysis and the underlying physical arguments behind these choices are presented. The expressions for the m-KFVS fluxes are derived. For one of the choices, the expressions for the split fluxes are similar to the usual first order KFVS method. The mathematical properties of 1D m-KFVS fluxes and the eigenvalues of the corresponding flux Jacobians are studied numerically. The analysis of numerical dissipation is carried out both at Boltzmann and Euler levels. The expression for stability criterion is derived. In order to be consistent with the interior scheme, modified solid wall and outer boundary conditions are derived by extending the MCIR idea to boundaries. The cell-centred finite volume method based on m-KFVS is applied to several standard test cases for 1D, 2D and 3D inviscid flows. In the case of subsonic flows, the m-KFVS method produces much less numerical entropy compared to first order KFVS method and the results are comparable to second order accurate q-KFVS method. In transonic and supersonic flows, m-KFVS generates much less numerical dissipation compared to first order KFVS and even less compared to q-KFVS method. Further, the m-KFVS method captures the discontinuities more sharply with contours being smooth and near second order accuracy has been achieved in smooth regions, by still using first order stencil. Therefore, the numerical dissipation generated by m-KFVS is considerably reduced by suitably choosing the dissipation control variables. The Euler code based on m-KFVS method almost takes the same amount of computational time as that of KFVS method. Although, the formal accuracy is of order one, the m-KFVS method resolves the flow features much more accurately compared to first order KFVS method but the numerical dissipation generated by m-KFVS method may not be minimal. Hence, the dissipation control vector is in general not optimal. If we can find the optimal dissipation control vector then we will be able to achieve the minimal dissipation. In the present work, the above objective is attained by posing the minimisation of numerical dissipation in m-KFVS method as an optimal control problem. Here, the control variables are the dissipation control vector. The discrete form of the cost function, which is to be minimised is considered as the sum of the squares of change in entropy at all cells in the computational domain. The number of control variables is equal to the total number of cells or finite volumes in the computational domain, as each cell has only one dissipation control variable. In the present work, the minimum value of cost function is obtained by using gradient based optimisation method. The sensitivity gradients of the cost function with respect to the control variables are obtained using discrete adjoint approach. The discrete adjoint equations for the optimisation problem of minimising the numerical dissipation in m-KFVS method applied to 2D and 3D Euler equations are derived. The method of steepest descent is used to update the control variables. The automatic differentiation tool Tapenade has been used to ease the development of adjoint codes. The m-KFVS code combined with discrete adjoint code is applied to several standard test cases for inviscid flows. The test cases considered are, low Mach number flows past NACA 0012 airfoil and two element Williams airfoil, transonic and supersonic flows past NACA 0012 airfoil and finally, transonic flow past Onera M6 wing. Numerical results have shown that the m-KFVS-adjoint method produces even less numerical dissipation compared to m-KFVS method and hence results in more accurate solution. The m-KFVS-adjoint code takes more computational time compared to m-KFVS code. The present work demonstrates that it is possible to achieve near second order accuracy by formally first order accurate m-KFVS scheme while retaining advantages of first order accurate methods.

In the thesis work, a code to solve the two dimensional compressible Euler equations for external flows around arbitrary geometries have been developed. A Cartesianmesh generator is incorporated to the solver. Hence the pre-processing can be performed together with the solution within a single code. The code is written in the C++ programming language and its object oriented capabilities have been exploited to save memory in the data structure developed. The Cartesian mesh is formed by dividing squares successively into its four quadrants. The main advantage of using this type of a mesh is the ability to generate meshes around geometries of arbitrary complexity quickly and to adapt the mesh easily based on the solution. The main disadvantage of this method is that the treatment of the cells that are cut by the geometry. For the solution procedure Roe&rsquo / s method as well as flux vector splitting methods are used for the flux evaluation. The flux vector splitting schemes used are van Leer, AUSM, AUSMD and AUSMV methods. Time discretization is performed using a multi-stage method. To increase the accuracy least squares reconstruction is employed. The code is validated by performing calculations around a NACA0012 airfoil profile. The effect of reconstruction is demonstrated by plotting the pressure coefficient on the airfoil. The distribution obtained using reconstruction is very close to the experimental one while there is a considerable deviation for the case without reconstruction. Also the shock capturing capabilities of different methods have been investigated. In addition the performance of each method is analyzed for flow around an NLR 7301 airfoil with a flap.

The aim of this study is to develop a code that is capable of solving three-dimensional compressible flows which are viscous and turbulent, and parallelization of this code. Purpose of parallelization is to obtain a computational efficiency in time respect which enables the solution of complex flow problems in reasonable computational times. In the first part of the study, which is the development of a three-dimensional Navier-Stokes solver for turbulent flows, first step is to develop a two-dimensional Euler code using Roe flux difference splitting method. This is followed by addition of sub programs involving calculation of viscous fluxes. Third step involves implementation of Baldwin-Lomax turbulence model to the code. Finally, the Euler code is generalized to three-dimensions. At every step, code validation is done by comparing numerical results with theoretical, experimental or other numerical results, and adequate consistency between these results is obtained. In the second part, which is the parallelization of the developed code, two-dimensional code is parallelized by using Message Passing Interface (MPI), and important improvements in computational times are obtained.

A fully automated Cartesian/Quad grid generator and laminar flow solver have been developed for external flows by using C++. After defining the input geometry by nodal points, adaptively refined Cartesian grids are generated automatically. Quadtree data structure is used in order to connect the Cartesian cells to each other. In order to simulate viscous flows, body-fitted quad cells can be generated optionally. Connectivity is provided by cut and split cells such that the intersection points of Cartesian cells are used as the corners of quads at the outmost row. Geometry based adaptation methods for cut, split cells and highly curved regions are applied to the uniform mesh generated around the geometry. After obtaining a sufficient resolution in the domain, the solution is achieved with cellcentered approach by using multistage time stepping scheme. Solution based grid adaptations are carried out during the execution of the program in order to refine the regions with high gradients and obtain sufficient resolution in these regions. Moreover, multigrid technique is implemented to accelerate the convergence time significantly. Some tests are performed in order to verify and validate the accuracy and efficiency of the code for inviscid and laminar flows.

Least Square Kinetic Upwind Method (LSKUM) belongs to the class of mesh-less method that solves compressible Euler equations of gas dynamics. LSKUM is kinetic theory based upwind scheme that operates on any cloud of points. Euler equations are derived from Boltzmann equation (of kinetic theory of gases) after taking suitable moments. The basic update scheme is formulated at Boltzmann level and mapped to Euler level by suitable moments. Mesh-less solvers need only cloud of points to solve the governing equations. For a complex configuration, with such a solver, one can generate a separate cloud of points around each component, which adequately resolves the geometric features, and then combine all the individual clouds to get one set of points on which the solver directly operates. An obvious advantage of this approach is that any incremental changes in geometry will require only regeneration of the small cloud of points where changes have occurred. Additionally blanking and de-blanking strategy along with overlay point cloud can be adapted in some applications like store separation to avoid regeneration of points. Naturally, the mesh-less solvers have advantage in tackling complex geometries and moving components over solvers that need grids. Conventionally, higher order accuracy for space derivative term is achieved by two step defect correction formula which is computationally expensive. The present solver uses low dissipation single step modified CIR (MCIR) scheme which is similar to first order LSKUM formulation and provides spatial accuracy closer to second order. The maximum time step taken to march solution in time is limited by stability criteria in case of explicit time integration procedure. Because of this, explicit scheme takes a large number of iterations to achieve convergence. The popular explicit time integration schemes like four stages Runge-Kutta (RK4) are slow in convergence due to this reason. The above problem can be overcome by using the implicit time integration procedure. The implicit schemes are unconditionally stable i.e. very large time steps can be used to accelerate the convergence. Also it offers superior robustness. The implicit Lower-Upper Symmetric Gauss-Seidel (LU-SGS) scheme is very attractive due to its low numerical complexity, moderate memory requirement and unconditional stability for linear wave equation. Also this scheme is more efficient than explicit counterparts and can be implemented easily on parallel computers. It is based on the factorization of the implicit operator into three parts namely lower triangular matrix, upper triangular matrix and diagonal terms. The use of LU-SGS results in a matrix free implicit framework which is very economical as against other expensive procedures which necessarily involve matrix inversion. With implementation of the implicit LU-SGS scheme larger time steps can be used which in turn will reduce the computational time substantially. LU-SGS has been used widely for many Finite Volume Method based solvers. The split flux Jacobian formulation as proposed by Jameson is most widely used to make implicit procedure diagonally dominant. But this procedure when applied to mesh-less solvers leads to block diagonal matrix which again requires expensive inversion. In the present work LU-SGS procedure is adopted for mesh-less approach to retain diagonal dominancy and implemented in 2-D and 3-D solvers in matrix free framework. In order to assess the efficacy of the implicit procedure, both explicit and implicit 2-D solvers are tested on NACA 0012 airfoil for various flow conditions in subsonic and transonic regime. To study the performance of the solvers on different point distributions two types of the cloud of points, one unstructured distribution (4074 points) and another structured distribution (9600 points) have been used. The computed 2-D results are validated against NASA experimental data and AGARD test case. The density residual and lift coefficient convergence history is presented in detail. The maximum speed up obtained by use of implicit procedure as compared to explicit one is close to 6 and 14 for unstructured and structured point distributions respectively. The transonic flow over ONERA M6 wing is a classic test case for CFD validation because of simple geometry and complex flow. It has sweep angle of 30° and 15.6° at leading edge and trailing edge respectively. The taper ratio and aspect ratio of the wing are 0.562 and 3.8 respectively. At M∞=0.84 and α=3.06° lambda shock appear on the upper surface of the wing. 3¬D explicit and implicit solvers are tested on ONERA M6 wing. The computed pressure coefficients are compared with experiments at section of 20%, 44%, 65%, 80%, 90% and 95% of span length. The computed results are found to match very well with experiments. The speed up obtained from implicit procedure is over 7 for ONERA M6 wing. The determination of the aerodynamic characteristics of a wing with the control surface deflection is one of the most important and challenging task in aircraft design and development. Many military aircraft use some form of the delta wing. To demonstrate the effectiveness of 3-D solver in handling control surfaces and small gaps, implicit 3-D code is used to compute flow past clipped delta wing with aileron deflection of 6° at M∞ = 0.9 and α = 1° and 3°. The leading edge backward sweep is 50.4°. The aileron is hinged from 56.5% semi-span to 82.9% of semi-span and at 80% of the local chord from leading edge. The computed results are validated with NASA experiments

The present work deals with the extension of Acoustic Flux Vector Splitting (AFVS) scheme for the Compressible Viscous flow computations. Accurate viscous flow computations require much finer grids with adequate clustering of grid points in certain regions. Viscous flow computations are performed on unstructured triangulated grids. Solving Navier-Stokes equations involves the inviscid Euler part and the viscous part. The inviscid part of the fluxes are computed using the Acoustic Flux Vector Splitting scheme and the viscous part which is diffusive in nature does not require upwinding and is taken care using a central difference type of scheme. For these computations both the cell centered and the cell vertex finite volume methods are used. Higher order accuracy on unstructured meshes is achieved using the reconstruction procedure. Test cases are chosen in such a way that the performance of the scheme can be evaluated for different range of mach numbers. We demonstrate that higher order AFVS scheme in conjunction with a suitable grid adaptation strategy produce results that compare well with other well known schemes and the experimental data. An assessment of the relative performance of the AFVS scheme with the Roe scheme is also presented.

Les phénomènes de transport en milieux poreux sont étudiés depuis près de deux siècles, cependant les travaux concernant les milieux fortement poreux sont encore relativement peu nombreux. Les modèles couramment utilisés pour les poreux classiques (lits de grains par exemple) sont peu applicables pour les milieux fortement poreux (les mousses par exemple), un certain nombre d’études ont été entreprises pour combler ce manque. Néanmoins, les résultats expérimentaux et numériques caractérisant les pertes de charge dans les mousses sont fortement dispersés. Du fait des progrès de l’imagerie 3D, une tendance émergente est la détermination des paramètres des lois d’écoulement à partir de simulations directes sur des géométries reconstruites. Nous présentons ici l’utilisation d’une nouvelle approche cinétique pour résoudre localement les équations de Navier-Stokes et déterminer les propriétés d’écoulement (perméabilité, dispersion, ...). / A novel kinetic scheme satisfying an entropy condition is developed, tested and implemented for the simulation of practical problems. The construction of this new entropic scheme is presented. A classical hyperbolic system is approximated by a discrete velocity vector kinetic scheme (with the simplified BGK collisional operator), but applied to an inviscid compressible gas dynamics system with a small Mach number parameter, according to the approach of Carfora and Natalini (2008). The numerical viscosity is controlled, and tends to the physical viscosity of the Navier-Stokes system. The proposed numerical scheme is analyzed and formulated as an explicit finite volume flux vector splitting (FVS) scheme that is very easy to implement. It is close in spirit to Lattice Boltzmann schemes, but it has the advantage to satisfy a discrete entropy inequality under a CFL condition and a subcharacteristic stability condition involving a cell Reynolds number. The new scheme is proved to be second-order accurate in space. We show the efficiency of the method in terms of accuracy and robustness on a variety of classical benchmark tests. Some physical problems have been studied in order to show the usefulness of both schemes. The LB code was successfully used to determine the longitudinal dispersion of metallic foams, with the use of a novel indicator. The entropic code was used to determine the permeability tensor of various porous media, from the Fontainebleau sandstone (low porosity) to a redwood tree sample (high porosity). These results are pretty accurate. Finally, the entropic framework is applied to the advection-diffusion equation as a passive scalar.