Landmarks And New Frontiers Of Computational Fluid Dynamics
Shang Advances in Aerodynamics(2019) Advances in AerodynamicsOpen AccessLandmarks and new frontiers ofcomputational fluid dynamicsJoseph J. S. ShangCorrespondence: joseph.shang@wright.eduWright State University, 3640Colonel Glenn Highway, Dayton,OH 45435-0001, USAAbstractA narrative of landmarks in computational fluid dynamics (CFD) is presented tohighlight the cornerstone achievements. Illuminating accomplishments starting fromthe very beginning of the coherent development until the most recent progress willbe elucidated over the span over more than six decades. Meanwhile, the cuttingedge scientific innovations will also be discussed for their lasting impacts to fluiddynamics and the physics-based modeling and simulation discipline. To traversesuch a vast domain over time by a single presentation, numerous and excellentcontributions to CFD will be unavoidably overlooked. Nevertheless it is my ardenthope that the present discussion will be able to reaffirm excellence in research andto identify new frontiers for scientific research. Especially, the challenges to futureinnovations will also be delineated to recommend for potential and fertile researchareas for the modeling and simulation science.Keywords: Computational fluid dynamics, Numerical algorithms, Turbulence,Interdisciplinary computational fluid dynamics1 IntroductionIn order to discuss the physics-based modeling and simulation discipline, the underlying principles must be explicitly stipulated to define its limitations. The traditionalcomputational fluid dynamics (CFD) technique is mostly applying in the continuumgas domain which is limited to the negligible Knudson number; Nn λ/l 1.0. In thisphysical domain, the mean-free-path of particle collisions is negligible in comparisonwith the characteristic length of the flowfield considered. In the continuum regime, thecompressible Navier-Stokes equations become the governing equation for describingfluid dynamics in the macroscopic scale, and the nonlinear partial differential equations system is the incompletely parabolic type. Even though the incompressibleNavier-Stokes was known to us as far back as 1827, only more recently the system ofequations was derived in integral form via the control-volume formulation to becomethe basis for the finite-volume approach. Nevertheless, the necessary initial values,boundary conditions, and their placement and implementation are mandatory toachieve a unique numerical simulation.Numerical algorithms are inseparable parts of CFD research, also are the most demanding and creative efforts of this discipline, because they dictate the computationalaccuracy that provides the required physical fidelity to any computational simulations.In the gist, the numerical algorithm and the computational procedure research is a tool The Author(s). 2019 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 InternationalLicense (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium,provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, andindicate if changes were made.
Shang Advances in Aerodynamics(2019) 1:5development endeavor. The type of research is subjected to the most rigorous scrutinizing for consistency, uniqueness, and stability issues in numerical analysis. For this reason, the value of a numerical algorithm always rises and falls according to practicalproblem solving needs. The adopted numerical algorithms at the early stage of CFD development are mostly explicit schemes for their simplicity in programming and limitingby the computer memory. The widen CFD application demands have driven computertechnology from scalar to vector and finally to concurrent data processing, meanwhilethe unconditionally stable implicit schemes also reached maturate. When CFD expanded to all flow regimes from subsonic, transonic, supersonic, to hypersonic; theneed for treating piecewise continuous numerical solutions is paramount and the levelof sophistication also elevated. There are simply too many numerical algorisms to becompletely and precisely discussed, as the consequence, only the classic results thathave withheld the test of time are included together with the most recent progress inhigh-resolution procedures.The landmarks for CFD accomplishments are presented according to their contributions to scientific discovery in fluid dynamics and technical breakthrough to aerospaceengineering. The brief review starts from a very few fundamental concepts which leadto a coherent development during WWII in the middle of 1940s. Followed a remarkable growing period initiate by NASA (National Aeronautics and Space Administration)research centers in the later 1960s, the pursuit of scientific excellence was firmly instilled in this technical discipline. In the subsequence years, CFD expands into interdisciplinary arenas for combustion, propulsion, structure dynamics, flight control, thermalprotection for earth reentry space vehicles. The matured technology has opened avenues for aerospace vehicle design and analysis; it is self-evident that CFD was widelyused for the Space Shuttle design and evaluation, and the National Aerospace Plane(NASP) was entirely designed by CFD techniques. Shortly afterwards, the CFD techniques are transferring into the computational electromagnetics and computationalmagnetohydrodynamics disciplines.The arriving of concurrent, high performance super computational technology provides an extraordinary opportunity for CFD to create many new science frontiers. Thefirst and the straightforward opportunities are to address the most challenged and theleast understood fluid dynamics phenomena such as the bifurcation, hysteresis, and turbulence. Based on the kinetic theory of gas, these fluid dynamic phenomena are addressable by direct numerical simulation with accurate initial values and boundaryconditions, without imposing any statistical ensemble approximations. The second andgreater challenge is expanding the scientific basis for simulating high enthalpy or hightemperature gas phenomenon by removing the elastic collision restrictions from kinetictheory of gas; namely the internal degree of freedom in vibrational and electron excitations of atom and molecule will be described by inelastic collisions involving quantummechanics. Based on our accumulated knowledge, these opportunities and possiblenew approaches will be discussed and outlined.2 Governing equationsThe governing equation for traditional CFD in continuum domain is the time-dependent,compressible Navier-Stokes equations, which first published in 1827 for incompressibleflows [1]. The closure of the nonlinear partial differential equations system was achievedPage 2 of 36
Shang Advances in Aerodynamics(2019) 1:5Page 3 of 36by Stokes through the relationship between the bulk and molecular viscosity coefficients[2]. In the strict sense, the governing equations are applicable only to the Newtonian fluidfor which the shearing stress is linearly proportional to the rate of strain. In essence, theNavier-Stokes equations describe gas particle dynamics on the macroscopic scale. Thesystem of equations is germinated from the kinetic theory of gas [3]: Within a givendynamic system, the gas particles always move in random motion with the kinetic andpotential energy of their own. The individual particle’s behavior in microscopic scales canonly be meaningfully described through statistic means. Based on the probability theory,it introduces a weighing factor known as the distribution function, f(xi,ci,t) in thesix-degree-of-freedom geometric and velocity space or the phase space. By the definitionof Hamiltonian that a system of particle possesses kinetic and potential energy, the ratesof change have the symmetric property relating to the particle velocity and geometricposition in the phase space. According to the Liouville’s theorem, the number density of adynamic system of moving particles in the phase space must remain constant. The rate ofchanges for the distribution function of the particles is governing by the Boltzmann orBoltzmann-Maxwell equation of the distribution function, f ðxi ; ci ;t Þ t þ ci f ðxi ; ci ;t Þ þ F i u f ðxi ; ci ;t Þ ¼ ½ f ðxi ; ci ;t Þ t cð1Þwhere ci and xi are the specular velocity and the position of particles in the phasespace. The external force exerting on each particle is designated as Fi. Theintegro-differential equation is very difficult to solve, and the solutions of the Boltzmann equation in term of probability is also not suitable for engineering applications.In order to simplify the Boltzmann equation, the particles dynamics contributed bythe collision integral on the right-hand-side of the Maxwell equation are simplified by agroup of elastic spheres. This simplification makes the total energy of particle’s internaldegree of freedom an invariant; in other words, the internal excitations of the particlesare neglected. As the consequence, the inter-atomic and inter-molecular excitationsand energy cascading between internal modes of gas particles are not considered. In anaddition, the interaction of particles is limited to binary encounters. The binary dynamic exchange by elastic collision actually establishes the concept of collision equilibrium condition, leading to:h 0 0 i 0 ½ f ðxi ; ci ; t Þ t c ¼ f ci f xi f ðci Þ f ðxi Þ ci ci d 3 xi d 3 cið2ÞUnder the dynamic equilibrium condition; 0 0 f ci f xi ¼ f ðci Þ f ðxi Þð3ÞThe link between the microscopic and macroscopic description of gasdynamics canbe established by the method of moments, but the most successful approach is theEnskog’s infinite series expansion [3]. Under the collision equilibrium condition, theBoltzmann equations transform directly to the Euler equations, which are essentiallythe Navier-Stokes equations but containing only the inviscid terms. The hierarchy offluid dynamics governing equation is depicting in Fig. 1.Under the nonequilibrium collision condition, the transport properties of the gasmust be included. To be consistent with theoretic formulation, the transport propertiesof gas are obtained by the gas kinetic theory of diluted gas mixtures, and it is a
Shang Advances in Aerodynamics(2019) 1:5Page 4 of 36Fig. 1 The hierarchy of conservation lawslandmark achievement by the kinetic theory of gas [4]. The transport property of anycombination of gaseous mixture is derivable by the inter-molecular potential function.The required collision integrals and cross sections for the gas molecular viscosity,thermal conductivity, and binary diffusion coefficients of individual species have beenobtained by the Lenard-Jones potential for gas molecules [5].The diffusion coefficient of a binary gas mixture is;Di; j ¼ 1:858 10 �ffiffiffiffiffiffiffiffiffi T 3 M i þ M j M i M j σ 2i; j Ωð1;1Þð4 aÞThe molecular viscosity of a single species is given as.μ ¼ 2:67 10 5pffiffiffiffiffiffiffiffiffiffi 2 ð2;2ÞM i T σ i Ωð4 bÞand the thermal conductivities for a mono-atomic and poly-atomic molecules are;pffiffiffiffiffiffiffiffiffiffiffiffi 2 ð2;2ÞT M i σ i Ωð4 cÞpffiffiffiffiffiffiffiffiffiffiffiffi 2 ð2;2ÞT M i σ i Ωð4 dÞκ i;m ¼ 1:989 10 4κ i;p ¼ 2:519 10 4The collision integrals Ω(1, 1), Ω(2, 2) and the transport cross section σi are obtained byperforming three consecutive integrations of the inter-molecular potential function:The integrations are performed first to determine the classic deflection angle as theimpact parameter, then from the impact parameters to get the relevant cross section.Finally, an averaging process is carried over the entire range of energy to produce thecollision cross section as a function of temperature [6].For an inhomogeneous gas mixture, the transport property can be approximated bythe Wilke’s mixing rule [5]. However, in most practical engineering applications, thetransport properties of air by kinetic theory of gas are replaced by empirical formulations and similarity aerodynamic parameters such as the Prandtl number. Whence thetransport properties are known, the time-dependent, three-dimensional, compressibleNavier-Stokes equations can be given as;
Shang Advances in Aerodynamics(2019) 1:5Page 5 of 36 ρþ ðρuÞ ¼ 0 tð5 aÞ ρuþ ðρuu τ Þ ρf ¼ 0 tð5 bÞ ρeþ ðρeu þ q þ u τ Þ ρð f uÞ ¼ 0 tð5 cÞThe system of equations is also known as the conservation laws. Equation (5-a) is oftenreferred to as the continuity equation which is the most fundamental concept of Newtonianmechanics in that the mass and energy is not exchangeable like in quantum mechanics.The conservation of momentum equation, Eq. (5-b) is the only vector equation in thesystem, and it is the Newton’s second law of motion with a possible external force f, such asgravitation or the electromagnetic force. The nonlinear transfer of momentum by convection is represented by a dyadic, ρuu, which is the principal component of the inviscid terms.In fact, it is also the source of turbulence from vortex interactions within the entire flowfield.The shear stress term, on the other hand, is another second rank tensor described as;τ ¼ ð p þ λ uÞI þ μdef ðuÞð6 aÞwhere, λ and μ are the bulk and molecular viscosity and I is the identity matrix. Thelast term of the stress tensor is referred to as the deformation tensor;def ðuÞ ¼ u þ ð uÞTð6 bÞThe transpose operator of the gradient u, ( u)T is simply by replacing the rows bycolumns in the matrix element of u. The deformation tensor has an important fluidmechanical interpretation in that the diagonal derivatives represent the longitudinalstrain, while the off-diagonal derivatives represent the angular deformation of fluid motion. As the consequences, the viscous flow at the solid-fluid interface boundary producesshear stress. Whereas, the inviscid terms associated with the normal component of thestress tensor lead to expansion and compression of the flow.Equation (5-c) is the conservation of energy law, it is just the second law of thermodynamics, and the internal energy is defined as; Z ð6 cÞρe ¼ ρcc dT þ u u 2The heat transfer term includes the Fourier’s law for conductive, convective by different species with different diffusion velocity, and the radiation energy transfer;q ¼ k T þ Σρi ui hi þ qradð6 dÞIt is important to know that the system of equations, Eq. (5-a), (5-b), and (5-c) constitute a nonlinear, incompletely parabolic partial differential equation system [7]. Anyunique solutions to the compressible Navier-Stokes equation must satisfy the compatibleinitial values and boundary conditions to the differential equations system.It is interesting to realize that the widely adopted finite-volume formulation ofNavier-Stokes equations in integral form via a control volume formulation was firstformerly derived by Rizzi and Inouye [8]. The balancing of outward normal vector flux
Shang Advances in Aerodynamics(2019) 1:5Page 6 of 36components across the control surface between adjacent control volumes becomes theonly constraint to the flux vector splitting technique.In practical applications, the conservation laws Eq.; (5-a), (5-b), and (5-c) are usuallywritten in a strong conservation flux vector form. On the Cartesian coordinates, theyappear as U t þ F x þ G y þ H z ¼ 0ð7 aÞwhere the dependent variables are;U U(ρ, ρu, ρv, ρw, ρe).The Rankine-Hugoniot jump condition across a shock wave is recoverable from theEuler equations which constitutes the hyperbolic partial differential equations. Therefore, the conservation laws are often solved separately but concurrently for the inviscidand viscous terms. For this reason, the flux vectors F, G, and H are often split into components of inviscid and viscous terms as; U t þ ð F i þ F v Þ x þ ðGi þ Gv Þ y þ ðH i þ H v Þ z ¼ 0ð7 bÞFor simulating complex configurations, the flux vector formulation is often transformed onto a generalized curvilinear, body oriented coordinate by means of the chainrule of differentiation [9]. Again through metric identities of coordinate transformation,the equation can still be rewritten in the strong conservation form.3 Numerical algorithms evolutionThe numerical algorithm is the heart of computational fluid dynamic, because it is the necessary translator between numerical analysis for fluid dynamics via computers. The historyof CFD is also ultimately related to the development of programmable digital computers: In1833 Charles Baggage originated the idea of a programmable computer, but the first patentfor the ENIAC computer (electronic numerical integrator and computer) was recorded in1947 for the truly programmable computer using transistors.The interrelation between numerical algorithms and computational results is depictedby a graphic presentation in Fig. 2. The illustrated algorithm is the diminishing residuereturn (DRR) scheme; the right-hand-side of the conservative law represents the physicsFig. 2 Relationship between physics fidelity and numerical algorithm
Shang Advances in Aerodynamics(2019) 1:5to be simulated. The left-hand-side of the equation is the numerical process and its solepurpose is keeping a stable computation. In fact, the illustration also implies the equivalent principle held for which if a stable numerical algorithm leading to a convergedasymptote, the numerical result is ensured to be the unique solution.There are two entirely different concepts for CFD formulations, and the most widelyadopted approach is the Eulerian frame of reference. In this formulation the fluiddynamics is analyzed in a control volume fixed in space. Whereas, the Lagrangianapproach is analyzing fluid dynamics by following a group moving gas particles in anenclosed control volume. The well-known direct simulation Monte Carlo (DSMC)method is built on the Largangian formulation, together with the particle-in-cell (PIC)method by Harlow [10]. For the PIC method, the fluid dynamics is represented byLagrangian mass particles within a control volume. At each time step, the calculatedinternal energy and velocity are obtained and the conservation properties are checkedby the sum of these final values before the process advances to the next time level. TheDSMC and PIC methods have demonstrated to be well suited to study the timedependent and multi-spices fluid medium, and had been widely used for simulatingrarefied gasdynamics and plasma dynamics [11].The most predominant CFD algorithm pioneers are led by Richardson who introduced point iterative scheme to solve the elliptic partial differential equation as far backas 1910 [12]. Then Courant, Friedrichs and Lewy initiated the rigorous investigationprocedure for examining the stability of a numerical algorithm by Fourier analysis in1928. They also addressed the uniqueness and existence of the numerical results forpartial differential equations [13]. It was Southwell who introduced a relaxation schemeto solve both the fluid dynamic and structure problem to become an accepted procedure for engineering application in 1940 [14]. Lax [15] and Godunov [16] addressed themost challenging and difficult issues in numerical analyses for resolving discontinuousfluid phenomena in a discrete space – the approximate Riemann problem. As it will beseen later, it remains to be the most studied problem in CFD.In the early 1960s, the dominated numerical algorithms are mostly explicit schemes,such as the Lax-Wendroff, leap-frog, and fractional step methods for multi-dimensionalproblems [17]. When CFD ventures into increasingly complex fluid phenomena, the moreefficient and stable implicit schemes are required. Especially, the ADI method [18, 19] hasbeen effectively applied to all type of partial equations, except when applying to thetime-dependent, three-dimensional hyperbolic system for which some forms of artificialdissipative terms must be appe
Keywords: Computational fluid dynamics, Numerical algorithms, Turbulence, Interdisciplinary computational fluid dynamics 1 Introduction In order to discuss the physics-based modeling and simulation discipline, the under-lying principl
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