Improved upwind discretization of the advection equation

2013 ◽  
Vol 30 (3) ◽  
pp. 773-787 ◽  
Author(s):  
Utku Erdogan
Keyword(s):  
Advection Equation ◽  
Upwind Discretization

scholarly journals Detailed Modeling of Cork-Phenolic Ablators in Preparation for the Post-flight Analysis of the QARMAN Re-entry CubeSat

2021 ◽  
Author(s):  
Claudio Miccoli ◽  
Alessandro Turchi ◽  
Pierre Schrooyen ◽  
Domenic D’Ambrosio ◽  
Thierry Magin
Keyword(s):  
Fluid Dynamics ◽  
Thermal Protection ◽  
A Priori ◽  
European Space Agency ◽  
Thermal Response ◽  
Accurate Method ◽  
Navier Stokes ◽  
Advection Equation ◽  
High Enthalpy ◽  
Space Agency

AbstractThis work deals with the analysis of the cork P50, an ablative thermal protection material (TPM) used for the heat shield of the qarman Re-entry CubeSat. Developed for the European Space Agency (ESA) at the von Karman Institute (VKI) for Fluid Dynamics, qarman is a scientific demonstrator for Aerothermodynamic Research. The ability to model and predict the atypical behavior of the new cork-based materials is considered a critical research topic. Therefore, this work is motivated by the need to develop a numerical model able to respond to this demand, in preparation to the post-flight analysis of qarman. This study is focused on the main thermal response phenomena of the cork P50: pyrolysis and swelling. Pyrolysis was analyzed by means of the multi-physics Computational Fluid Dynamics (CFD) code argo, developed at Cenaero. Based on a unified flow-material solver, the Volume Averaged Navier–Stokes (VANS) equations were numerically solved to describe the interaction between a multi-species high enthalpy flow and a reactive porous medium, by means of a high-order Discontinuous Galerkin Method (DGM). Specifically, an accurate method to compute the pyrolysis production rate was implemented. The modeling of swelling was the most ambitious task, requiring the development of a physical model accounting for this phenomenon, for the purpose of a future implementation within argo. A 1D model was proposed, mainly based on an a priori assumption on the swelling velocity and the resolution of a nonlinear advection equation, by means of a Finite Difference Method (FDM). Once developed, the model was successfully tested through a matlab code, showing that the approach is promising and thus opening the way to further developments.

Eulerian partial-differential-equation methods for complex-valued eikonals in attenuating media

Geophysics ◽  
2021 ◽  
pp. 1-53
Author(s):  
Jiangtao Hu ◽  
Jianliang Qian ◽  
Jian Song ◽  
Min Ouyang ◽  
Junxing Cao ◽  
...  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Ray Tracing ◽  
Seismic Waves ◽  
Real Space ◽  
Advection Equation ◽  
Partial Differential ◽  
The Real ◽  
Eikonal Function ◽  
Complex Valued

Seismic waves in earth media usually undergo attenuation, causing energy losses and phase distortions. In the regime of high-frequency asymptotics, a complex-valued eikonal is an essential ingredient for describing wave propagation in attenuating media, where the real and imaginary parts of the eikonal function capture dispersion effects and amplitude attenuation of seismic waves, respectively. Conventionally, such a complex-valued eikonal is mainly computed either by tracing rays exactly in complex space or by tracing rays approximately in real space so that the resulting eikonal is distributed irregularly in real space. However, seismic data processing methods, such as prestack depth migration and tomography, usually require uniformly distributed complex-valued eikonals. Therefore, we propose a unified framework to Eulerianize several popular approximate real-space ray-tracing methods for complex-valued eikonals so that the real and imaginary parts of the eikonal function satisfy the classical real-space eikonal equation and a novel real-space advection equation, respectively, and we dub the resulting method the Eulerian partial-differential-equation method. We further develop highly efficient high-order methods to solve these two equations by using the factorization idea and the Lax-Friedrichs weighted essentially non-oscillatory (WENO) schemes. Numerical examples demonstrate that the proposed method yields highly accurate complex-valued eikonals, analogous to those from ray-tracing methods. The proposed methods can be useful for migration and tomography in attenuating media.

scholarly journals SOLUTION OF THE MULTIDIMENSIONAL DIFFUSION EQUATION USING LIE SYMMETRIES: SIMULATION OF POLLUTANT DISPERSION IN THE ATMOSPHERE

2005 ◽  
Vol 4 (2) ◽  
Author(s):  
J. R. Zabadal ◽  
C. A. Poffal
Keyword(s):  
Differential Operators ◽  
Hybrid Methods ◽  
Formal Solution ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Pollutant Dispersion ◽  
Form Solution ◽  
Lie Symmetries ◽  
Advection Equation ◽  
Mean Values

Several analytical, numerical and hybrid methods are being used to solve diffusion and diffusion advection problems. In this work, a closed form solution of the three-dimensional diffusion advection equation in a Cartesian coordinate system is obtained by applying rules, based on the Lie symmetries, to manipulate the exponential of the differential operators that appear in its formal solution. There are many advantages of applying these rules: the increase in processing velocity so that the solution may be obtained in real time, the reduction in the amount of memory required to perform the necessary tasks in order to obtain the solution, since the analytical expressions can be easily manipulated in post-processing and also the discretization of the domain may not be necessary in some cases, avoiding the use of mean values for some parameters involved. These rules yield good results when applied to obtain solutions for problems in fluid mechanics and in quantum mechanics. In order to show the performance of the method, a one-dimensional scenario of the pollutant dispersion in a stable boundary layer is simulated, considering that the horizontal component of the velocity field is dominant and constant, disregarding the other components. The results are compared with data available in the literature.

Simulation of Dendritic Growth Using a VOF Method

2009 ◽  
Author(s):  
Joaqui´n Lo´pez ◽  
Julio Herna´ndez ◽  
Claudio Zanzi ◽  
Fe´lix Faura ◽  
Pablo Go´mez
Keyword(s):  
Dendritic Growth ◽  
Piecewise Linear ◽  
Height Function ◽  
Vof Method ◽  
Three Dimensions ◽  
Advection Equation ◽  
Thermal Gradients ◽  
Interfacial Flows ◽  
Thermal Boundary Conditions ◽  
Interface Curvature

The volume of fluid (VOF) method is one of the most widely used methods to simulate interfacial flows using fixed grids. However, its application to phase change processes in solidification problems is relatively infrequent. In this work, preliminary results of the application of a new methodology to the simulation of dendritic growth of pure metals is presented. The proposed approach is based on a recent VOF method with PLIC (piecewise linear interface calculation) reconstruction of the interface. A diffused-interface method is used to solve the energy equation, which avoids the need of applying the thermal boundary conditions directly at the solid front. The thermal gradients at both sides of the interface, which are needed to accurately obtain the front velocity, are calculated with the aid of a distance function. The advection equation of a discretized solid fraction function is solved using the unsplit VOF advection method proposed by Lo´pez et al. [J. Comput. Phys. 195 (2004) 718–742] (extended to three dimensions by Herna´ndez et al. [Int. J. Numer. Methods Fluids 58 (2008) 897-921]). The interface curvature is computed using an improved height function (HF) technique, which provides second-order accuracy. The assessment of the proposed methodology is carried out by comparing the numerical results with analytical solutions and with results obtained by different authors for the formation of complex dendritic structures in two and three dimensions.

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