June  2019, 11(2): 187-203. doi: 10.3934/jgm.2019010

Conservation laws in discrete geometry

1. 

Computational Physics Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

2. 

Theoretical Design Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

* Corresponding author: L. G. Margolin

Received  April 2018 Revised  January 2019 Published  May 2019

The small length scales of the dissipative processes of physical viscosity and heat conduction are typically not resolved in the numerical simulation of high Reynolds number flows in the discrete geometry of computational grids. Historically, the simulations of flows with shocks and/or turbulence have relied on solving the Euler equations with dissipative regularization. In this paper, we begin by reviewing the regularization strategies used in shock wave calculations in both a Lagrangian and an Eulerian framework. We exhibit the essential similarities with Large Eddy Simulation models of turbulence, namely that almost all of these depend on the square of the size of the computational cell. In our principal result, we justify that dependence by deriving the evolution equations for a finite-sized volume of fluid. Those evolution equations, termed finite scale Navier-Stokes (FSNS), contain dissipative terms similar to the artificial viscosity first proposed by von Neumann and Richtmyer. We describe the properties of FSNS, provide a physical interpretation of the dissipative terms and show the connection to recent concepts in fluid dynamics, including inviscid dissipation and bi-velocity hydrodynamics.

Citation: Len G. Margolin, Roy S. Baty. Conservation laws in discrete geometry. Journal of Geometric Mechanics, 2019, 11 (2) : 187-203. doi: 10.3934/jgm.2019010
References:
[1]

A. Alexander, Duel at Dawn, Harvard University Press, Cambridge, MA, 2010.

[2]

H. Alsmeyer, Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam, J. Fluid Mech., 74 (1976), 497-513.

[3]

R. Becker, Stoßbwelle und detonation, (In German), Zeitschrift für Physik, 8 (1922), 321–362.

[4]

H. A. Bethe, On the theory of shock waves for an arbitrary equation of state, Classic Papers in Shock Compression Science, J.N. Johnson & R. Cheret, eds., Springer–Verlag, New York, 1998,421–492. doi: 10.1007/978-1-4612-2218-7_11.

[5]

S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. Math., 161 (2005), 223-342. doi: 10.4007/annals.2005.161.223.

[6]

J. P. Boris and D. L. Book, Flux–corrected transport, J. Comput. Phys., 11 (1973), 38-69. doi: 10.1006/jcph.1997.5756.

[7]

H. Brenner, Kinematics of volume transport, Physica A, 349 (2004), 11-59.

[8]

H. Brenner, Steady-state heat conduction in quiescent fluids: Incompleteness of the Navier–Stokes–Fourier equations, Physica A, 390 (2011), 3216-3244. doi: 10.1016/j.physa.2011.04.023.

[9]

E. J. CaramanaM. J. Shashkov and P. P. Whalen, Formulations of artificial viscosity for multi–dimensional shock wave computations, J. Comput. Phys., 144 (1998), 70-97. doi: 10.1006/jcph.1998.5989.

[10]

S. Y. ChenD. D. HolmL. G. Margolin and R. Zhang, Direct numerical simulations of the Navier–Stokes alpha model, Physica D, 133 (1999), 66-83. doi: 10.1016/S0167-2789(99)00099-8.

[11]

S. Y. ChenC. FoiasD. D. HolmE. OlsonE. S. Titi and S. Wynne, Camassa–Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett., 81 (1998), 5338-5341. doi: 10.1103/PhysRevLett.81.5338.

[12]

A. CheskidovD. D. HolmE. Olson and E. S. Titi, On a Leray-$\alpha$ model of turbulence, Proc. Royal Soc. A, 461 (2005), 629-649. doi: 10.1098/rspa.2004.1373.

[13]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer, NY, 2010, third edition. doi: 10.1007/978-3-642-04048-1.

[14]

R. J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal., 88 (1985), 223-270. doi: 10.1007/BF00752112.

[15]

L. Euler, Principes généraux du mouvement des fluides, Mém. Acad. Sci. Berlin, 11, 274–315. See also an English translation by T.E. Burton, 1999: “General laws of the motion of fluids,” Fluid Dyn., 34 (1999), 801–822.

[16]

G. L. Eyink, Energy dissipation without viscosity in ideal hydrodynamics, Physica D, 78 (1994), 222-240. doi: 10.1016/0167-2789(94)90117-1.

[17]

U. Frisch, Turbulence: The Legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge, 1995.

[18]

L. S. García–ColínR. M. Velasco and F. J. Uribe, Beyond the Navier–Stokes equations: Burnett hydrodynamics, Phys. Reports, 465 (2008), 149-189. doi: 10.1016/j.physrep.2008.04.010.

[19]

B. J. Geurts and D. D. Holm, Regularization modeling for large–eddy simulation, Phys. Fluids, 15 (2003), L13–L16. doi: 10.1063/1.1529180.

[20]

B. J. Geurts and D. D. Holm, Leray and LANS-$\alpha$ modeling of turbulent mixing, J. Turbulence, 7 (2006), Paper 10, 33 pp. doi: 10.1080/14685240500501601.

[21]

S. K. Godunov, Different Methods for Shock Waves, Moscow State University, (Ph.D. Dissertation), 1954.

[22]

C. J. Greenshields and J. M. Reese, The structure of shock waves as a test of Brenner's modifications to the Navier-Stokes equations, J. Fluid Mech., 580 (2007), 407-429. doi: 10.1017/S0022112007005575.

[23]

F. F. Grinstein, L. G. Margolin and W. J. Rider, Implicit Large Eddy Simulation, Cambridge University Press, NY, NY, 2007. doi: 10.1017/CBO9780511618604.

[24]

J. L. GuermondJ. T. Oden and S. Prudhomme, An interpretation of the Navier–Stokes alpha model as a frame–indifferent Leray regularization, Physica D, 177 (2003), 23-30. doi: 10.1016/S0167-2789(02)00748-0.

[25]

J. L. GuermondR. Pasquetti and B. Popov, Entropy viscosity method for nonlinear conservation laws, J. Comput. Phys., 230 (2011), 4248-4267. doi: 10.1016/j.jcp.2010.11.043.

[26]

A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 49 (1983), 357-393. doi: 10.1016/0021-9991(83)90136-5.

[27]

M. W. Hecht, D. D. Holm, M. R. Petersen and B. A. Wingate, The LANS-alpha and Leray turbulence parameterizations in primitive equation ocean modeling, J. Physics A, 41 (2008), 344009, 23 pp. doi: 10.1088/1751-8113/41/34/344009.

[28]

C. W. Hirt, Heuristic stability theory for finite difference equations, J. Comput. Phys., 2 (1968), 339-355.

[29]

D. D. Holm, Kármán–Howarth theorem for the Lagrangian–averaged Navier–Stokes–alpha model of turbulence, J. Fluid Mech., 467 (2002), 205-214. doi: 10.1017/S002211200200160X.

[30]

D. D. HolmJ. E. Marsden and T. S. Ratiu, Euler–Poincaré models of ideal fluids with nonlinear dispersion, Phys. Rev. Lett., 80 (1998), 4173-4176.

[31]

G.M. Kremer, An Introduction to the Boltzmann Equation and Transport Processes in Gases, Springer, NY, 2010. doi: 10.1007/978-3-642-11696-4.

[32]

P. D. Lax, Mathematics and physics, Bull. Amer. Math. Soc., 45 (2008), 135-152. doi: 10.1090/S0273-0979-07-01182-2.

[33]

J. Leray, Sur les movements dun fluide visqueux remplaissant lespace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.

[34] R. J. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.
[35]

L. G. Margolin, Finite-scale equations for compressible fluid flow, Phil. Trans. R. Soc. A, 367 (2009), 2861-2871. doi: 10.1098/rsta.2008.0290.

[36]

L. G. Margolin, The role of the observer in classical fluid flow, Mech. Res. Comm., 57 (2014), 10-17.

[37]

L. G. Margolin and A. Hunter, Discrete thermodynamics, Mech. Res. Comm., 93 (2018), 103-107. doi: 10.1016/j.mechrescom.2017.10.006.

[38]

L. G. Margolin and C. S. Plesko, Discrete regularization, Evolution Equations and Control Theory, 8 (2019), 117-137.

[39]

L. G. MargolinJ. M. Reisner and P. M. Jordan, Entropy in self-similar shock profiles, Int. J. Nonlinear Mech., 95 (2017), 333-346.

[40]

L. G. MargolinP. K. Smolarkiewicz and Z. Sorbjan, Large–eddy simulations of convective boundary layers using nonoscillatory differencing, Physica D., 133 (1999), 390-397. doi: 10.1016/S0167-2789(99)00083-4.

[41]

L. G. Margolin and W. J. Rider, A rationale for implicit turbulence modelling, Int. J. Num. Methods Fluids, 39 (2002), 821-841. doi: 10.1002/fld.331.

[42]

L. G. Margolin, W. J. Rider and F. F. Grinstein, Modeling turbulent flow with implicit LES, J. Turbulence, 7 (2006), Paper 15, 27 pp. doi: 10.1080/14685240500331595.

[43]

M. L. Merriam, Smoothing and the second law, Comp. Meth. Appl. Mech. Eng., 64 (1987), 177-193. doi: 10.1016/0045-7825(87)90039-9.

[44]

I. Múller, On the entropy inequality, Archive for Rational Mechanics and Analysis, 26 (1967), 118-141. doi: 10.1007/BF00285677.

[45]

P. Névir, Ertel's vorticity theorems, the particle relabeling symmetry and the energy–vorticity theory of mechanics, Meteorologische Zeitschrift, 13 (2004), 485-498.

[46]

W. F. Noh, Errors for calculations of strong shocks using an artificial viscosity and an artificial heat conduction, J. Comput. Phys., 72 (1978), 78-120.

[47]

E. S. Oran and J. P. Boris, Computing turbulent shear flows–a convenient conspiracy, Computers in Physics, 7 (1993), 523-533.

[48]

A. Petersen, The philosophy of Niels Bohr, Bulletin of the Atomic Scientists, 19 (1963), 8-14.

[49]

P. Saugat, Large Eddy Simulation for Incompressible Flows, Scientific Computation. Springer-Verlag, Berlin, 2006.

[50]

J. Smagorinsky, General circulation experiments with the primitive equations Ⅰ. The basic experiment, Mon. Wea. Rev., 91 (1963), 99-164.

[51]

B. Schmidt, Electron beam density measurements in shock waves in argon, J. Fluid Mech., 39 (1969), 361-373.

[52]

P. K. Smolarkiewicz and L. G. Margolin, MPDATA: A finite–difference solver for geophysical flows, J. Comput. Phys., 140 (1998), 459-480. doi: 10.1006/jcph.1998.5901.

[53]

G. G. Stokes, On the theories of the internal friction of fluids in motion, Trans. Camb. Phil. Soc., 8 (1845), 287-305.

[54]

P. A. Thompson, Compressible–Fluid Dynamics, McGraw–Hill, NY, 1972.

[55]

B. van Leer, Toward the ultimate conservative difference scheme V, J. Comput. Phys., 32 (1979), 101-136. doi: 10.1006/jcph.1997.5757.

[56]

J. von Neumann and R. D. Richtmyer, A method for the numerical calculation of hydrodynamic shocks, J. Appl. Phys., 21 (1950), 232-237. doi: 10.1063/1.1699639.

[57]

M. L. Wilkins, Use of artificial viscosity in multidimensional fluid dynamic calculations, J. Comput. Phys., 36 (1980), 281-303. doi: 10.1016/0021-9991(80)90161-8.

show all references

References:
[1]

A. Alexander, Duel at Dawn, Harvard University Press, Cambridge, MA, 2010.

[2]

H. Alsmeyer, Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam, J. Fluid Mech., 74 (1976), 497-513.

[3]

R. Becker, Stoßbwelle und detonation, (In German), Zeitschrift für Physik, 8 (1922), 321–362.

[4]

H. A. Bethe, On the theory of shock waves for an arbitrary equation of state, Classic Papers in Shock Compression Science, J.N. Johnson & R. Cheret, eds., Springer–Verlag, New York, 1998,421–492. doi: 10.1007/978-1-4612-2218-7_11.

[5]

S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. Math., 161 (2005), 223-342. doi: 10.4007/annals.2005.161.223.

[6]

J. P. Boris and D. L. Book, Flux–corrected transport, J. Comput. Phys., 11 (1973), 38-69. doi: 10.1006/jcph.1997.5756.

[7]

H. Brenner, Kinematics of volume transport, Physica A, 349 (2004), 11-59.

[8]

H. Brenner, Steady-state heat conduction in quiescent fluids: Incompleteness of the Navier–Stokes–Fourier equations, Physica A, 390 (2011), 3216-3244. doi: 10.1016/j.physa.2011.04.023.

[9]

E. J. CaramanaM. J. Shashkov and P. P. Whalen, Formulations of artificial viscosity for multi–dimensional shock wave computations, J. Comput. Phys., 144 (1998), 70-97. doi: 10.1006/jcph.1998.5989.

[10]

S. Y. ChenD. D. HolmL. G. Margolin and R. Zhang, Direct numerical simulations of the Navier–Stokes alpha model, Physica D, 133 (1999), 66-83. doi: 10.1016/S0167-2789(99)00099-8.

[11]

S. Y. ChenC. FoiasD. D. HolmE. OlsonE. S. Titi and S. Wynne, Camassa–Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett., 81 (1998), 5338-5341. doi: 10.1103/PhysRevLett.81.5338.

[12]

A. CheskidovD. D. HolmE. Olson and E. S. Titi, On a Leray-$\alpha$ model of turbulence, Proc. Royal Soc. A, 461 (2005), 629-649. doi: 10.1098/rspa.2004.1373.

[13]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer, NY, 2010, third edition. doi: 10.1007/978-3-642-04048-1.

[14]

R. J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal., 88 (1985), 223-270. doi: 10.1007/BF00752112.

[15]

L. Euler, Principes généraux du mouvement des fluides, Mém. Acad. Sci. Berlin, 11, 274–315. See also an English translation by T.E. Burton, 1999: “General laws of the motion of fluids,” Fluid Dyn., 34 (1999), 801–822.

[16]

G. L. Eyink, Energy dissipation without viscosity in ideal hydrodynamics, Physica D, 78 (1994), 222-240. doi: 10.1016/0167-2789(94)90117-1.

[17]

U. Frisch, Turbulence: The Legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge, 1995.

[18]

L. S. García–ColínR. M. Velasco and F. J. Uribe, Beyond the Navier–Stokes equations: Burnett hydrodynamics, Phys. Reports, 465 (2008), 149-189. doi: 10.1016/j.physrep.2008.04.010.

[19]

B. J. Geurts and D. D. Holm, Regularization modeling for large–eddy simulation, Phys. Fluids, 15 (2003), L13–L16. doi: 10.1063/1.1529180.

[20]

B. J. Geurts and D. D. Holm, Leray and LANS-$\alpha$ modeling of turbulent mixing, J. Turbulence, 7 (2006), Paper 10, 33 pp. doi: 10.1080/14685240500501601.

[21]

S. K. Godunov, Different Methods for Shock Waves, Moscow State University, (Ph.D. Dissertation), 1954.

[22]

C. J. Greenshields and J. M. Reese, The structure of shock waves as a test of Brenner's modifications to the Navier-Stokes equations, J. Fluid Mech., 580 (2007), 407-429. doi: 10.1017/S0022112007005575.

[23]

F. F. Grinstein, L. G. Margolin and W. J. Rider, Implicit Large Eddy Simulation, Cambridge University Press, NY, NY, 2007. doi: 10.1017/CBO9780511618604.

[24]

J. L. GuermondJ. T. Oden and S. Prudhomme, An interpretation of the Navier–Stokes alpha model as a frame–indifferent Leray regularization, Physica D, 177 (2003), 23-30. doi: 10.1016/S0167-2789(02)00748-0.

[25]

J. L. GuermondR. Pasquetti and B. Popov, Entropy viscosity method for nonlinear conservation laws, J. Comput. Phys., 230 (2011), 4248-4267. doi: 10.1016/j.jcp.2010.11.043.

[26]

A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 49 (1983), 357-393. doi: 10.1016/0021-9991(83)90136-5.

[27]

M. W. Hecht, D. D. Holm, M. R. Petersen and B. A. Wingate, The LANS-alpha and Leray turbulence parameterizations in primitive equation ocean modeling, J. Physics A, 41 (2008), 344009, 23 pp. doi: 10.1088/1751-8113/41/34/344009.

[28]

C. W. Hirt, Heuristic stability theory for finite difference equations, J. Comput. Phys., 2 (1968), 339-355.

[29]

D. D. Holm, Kármán–Howarth theorem for the Lagrangian–averaged Navier–Stokes–alpha model of turbulence, J. Fluid Mech., 467 (2002), 205-214. doi: 10.1017/S002211200200160X.

[30]

D. D. HolmJ. E. Marsden and T. S. Ratiu, Euler–Poincaré models of ideal fluids with nonlinear dispersion, Phys. Rev. Lett., 80 (1998), 4173-4176.

[31]

G.M. Kremer, An Introduction to the Boltzmann Equation and Transport Processes in Gases, Springer, NY, 2010. doi: 10.1007/978-3-642-11696-4.

[32]

P. D. Lax, Mathematics and physics, Bull. Amer. Math. Soc., 45 (2008), 135-152. doi: 10.1090/S0273-0979-07-01182-2.

[33]

J. Leray, Sur les movements dun fluide visqueux remplaissant lespace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.

[34] R. J. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.
[35]

L. G. Margolin, Finite-scale equations for compressible fluid flow, Phil. Trans. R. Soc. A, 367 (2009), 2861-2871. doi: 10.1098/rsta.2008.0290.

[36]

L. G. Margolin, The role of the observer in classical fluid flow, Mech. Res. Comm., 57 (2014), 10-17.

[37]

L. G. Margolin and A. Hunter, Discrete thermodynamics, Mech. Res. Comm., 93 (2018), 103-107. doi: 10.1016/j.mechrescom.2017.10.006.

[38]

L. G. Margolin and C. S. Plesko, Discrete regularization, Evolution Equations and Control Theory, 8 (2019), 117-137.

[39]

L. G. MargolinJ. M. Reisner and P. M. Jordan, Entropy in self-similar shock profiles, Int. J. Nonlinear Mech., 95 (2017), 333-346.

[40]

L. G. MargolinP. K. Smolarkiewicz and Z. Sorbjan, Large–eddy simulations of convective boundary layers using nonoscillatory differencing, Physica D., 133 (1999), 390-397. doi: 10.1016/S0167-2789(99)00083-4.

[41]

L. G. Margolin and W. J. Rider, A rationale for implicit turbulence modelling, Int. J. Num. Methods Fluids, 39 (2002), 821-841. doi: 10.1002/fld.331.

[42]

L. G. Margolin, W. J. Rider and F. F. Grinstein, Modeling turbulent flow with implicit LES, J. Turbulence, 7 (2006), Paper 15, 27 pp. doi: 10.1080/14685240500331595.

[43]

M. L. Merriam, Smoothing and the second law, Comp. Meth. Appl. Mech. Eng., 64 (1987), 177-193. doi: 10.1016/0045-7825(87)90039-9.

[44]

I. Múller, On the entropy inequality, Archive for Rational Mechanics and Analysis, 26 (1967), 118-141. doi: 10.1007/BF00285677.

[45]

P. Névir, Ertel's vorticity theorems, the particle relabeling symmetry and the energy–vorticity theory of mechanics, Meteorologische Zeitschrift, 13 (2004), 485-498.

[46]

W. F. Noh, Errors for calculations of strong shocks using an artificial viscosity and an artificial heat conduction, J. Comput. Phys., 72 (1978), 78-120.

[47]

E. S. Oran and J. P. Boris, Computing turbulent shear flows–a convenient conspiracy, Computers in Physics, 7 (1993), 523-533.

[48]

A. Petersen, The philosophy of Niels Bohr, Bulletin of the Atomic Scientists, 19 (1963), 8-14.

[49]

P. Saugat, Large Eddy Simulation for Incompressible Flows, Scientific Computation. Springer-Verlag, Berlin, 2006.

[50]

J. Smagorinsky, General circulation experiments with the primitive equations Ⅰ. The basic experiment, Mon. Wea. Rev., 91 (1963), 99-164.

[51]

B. Schmidt, Electron beam density measurements in shock waves in argon, J. Fluid Mech., 39 (1969), 361-373.

[52]

P. K. Smolarkiewicz and L. G. Margolin, MPDATA: A finite–difference solver for geophysical flows, J. Comput. Phys., 140 (1998), 459-480. doi: 10.1006/jcph.1998.5901.

[53]

G. G. Stokes, On the theories of the internal friction of fluids in motion, Trans. Camb. Phil. Soc., 8 (1845), 287-305.

[54]

P. A. Thompson, Compressible–Fluid Dynamics, McGraw–Hill, NY, 1972.

[55]

B. van Leer, Toward the ultimate conservative difference scheme V, J. Comput. Phys., 32 (1979), 101-136. doi: 10.1006/jcph.1997.5757.

[56]

J. von Neumann and R. D. Richtmyer, A method for the numerical calculation of hydrodynamic shocks, J. Appl. Phys., 21 (1950), 232-237. doi: 10.1063/1.1699639.

[57]

M. L. Wilkins, Use of artificial viscosity in multidimensional fluid dynamic calculations, J. Comput. Phys., 36 (1980), 281-303. doi: 10.1016/0021-9991(80)90161-8.

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