Magnetic Reynolds Number Effects in Compressible Magnetohydrodynamic Turbulence F

PHYSICS OF FLUIDS VOLUME 16, NUMBER 6 JUNE 2004

Magnetic Reynolds number effects in compressible magnetohydrodynamic turbulence F. Ladeinde Mechanical Engineering, State University of New York at Stony Brook, Stony Brook, New York 11794-2300 D. V. Gaitonde Air Vehicles Directorate, Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio 45433-7521 ͑Received 20 June 2003; accepted 10 March 2004; published online 4 May 2004͒

The effects of the magnetic Reynolds number, Re␴ , on decaying two-dimensional compressible magnetohydrodynamic ͑MHD͒ turbulence are investigated through direct numerical simulations. The initial relative intensities of, and the correlation between, the fluctuating velocity ͑u͒ and magnetic induction ͑b͒ fields are also varied, as measured with the respective parameters f and angle ␪. The investigations cover the parameter ranges 1рRe␴р250, 0°р␪р90°, and 1р f р3. The results suggest that, at the lowest Re␴ investigated, the magnetic field has a negligible impact on the evolution of the turbulence kinetic energy Ek . At higher Re␴ values, when magnetic effects are important, the magnetic field tends to accelerate the decay of the turbulence energy relative to non-MHD flows. On the other hand, the magnetic energy Eb shows the opposite trend, being rapidly driven from its initial values to essentially zero very early in the transient at lower Re␴ values, while higher Re␴ values significantly retard this decay. An enhancement of density fluctuations is noted in the intermediate Re␴ range. An interesting observation pertaining to the normalized cross helicity is the fast decay to zero of this quantity when Re␴ϭ1, independent of the values of ␪ and f. That is, the fluctuating u and b fields tend to be uncorrelated when the magnetic Reynolds number is low. In this case, the role of the magnetic field is passive, and it is merely convected by the velocity field. The conditions required to maintain a high correlation during the evolution are discussed. We have ␪ also seen that the Eb decay mode is less sensitive to the value of than that of Ek . The relative contribution of Ek , Eb , and the internal energy Ei to the total energy Et is discussed in relation to the values of f, ␪, and Re␴ .©2004 American Institute of Physics. ͓DOI: 10.1063/1.1736674͔

I. INTRODUCTION ϭ Ã the magnetic potential: b “ a. Previous work has also reported on spectral anisotropy that is dynamically generated The dynamics of electrically conducting fluid flows in in MHD turbulence when a strong mean magnetic field B0 is the presence of magnetic forces have been studied exten- present. For example, correlation lengths parallel to B0 were sively in fields as diverse as fusion physics,1–4 reported to be up to ten times the perpendicular scale in astrophysics,5–7 solar physics,8–10 and material processing.11 fusion-related machines.16,17 Also, Moffat.18 Sommeria and The present paper, however, is concerned with aerospace ap- Morcau,19 Davidson20,21 reported on the suppression of ho- plications of the subject,12,13 which is increasingly viewed as mogeneous turbulence at moderate to large values of the in- ϭ a pacing item for hypersonic flight. Although the knowledge teraction parameter, N Re␴ Rb , where Re␴ is the magnetic that electrical and magnetic forces can have a profound in- Reynolds number and Rb is the magnetic pressure number. fluence on aerospace-related flows is not new,14,15 the funda- Anisotropy, which is caused by the Lorentz force, has been mental study of magnetohydrodynamics ͑MHD͒ turbulence reported for both decaying and forced MHD turbulence, and in related canonical systems has not received enough atten- from theory, experiments, and numerical simulations. Zi- tion. The parameters encountered in high-speed naturally or kanov and Thess11 used direct numerical simulation ͑DNS͒ artificially ionized flows give rise to dynamics which are to study the transformation of initially isotropic turbulent expected to be considerably different from those that have flow of electrically conducting incompressible viscous fluid received attention in other fields. The behavior of the rugged under the influence of an imposed homogeneous magnetic Ӷ invariants that come into play in turbulent MHD flows, no- field. They considered magnetic Prandtl number Prm 1 and ␬ ϵ 1 " tably the mean cross helicity, Hc( ) 2͗u b͘, the mean assumed small values for the magnetic Reynolds, Re␴ ,a ␬ ϵ 1 " 22 magnetic helicity, Hm( ) 2͗a b͘, and mean electric current condition they used to justify the quasistatic approximation ␬ ϵ 1 " helicity, H j( ) 2͗j b͘, has been investigated in such non- and avoid the solution of the evolution equations for the aerospace fields as astrophysics. Here, angular brackets de- magnetic induction field. Large scale forcing was applied on note spatial averaging, ␬ is the magnitude of the wave num- ␬р2.5, where ␬ is the wave number, in order to maintain a ber vector, u, b, and j are the fluctuating components of the statistical steady state. The path of transformation, i.e., from velocity, magnetic, and current fields, respectively, and a is isotropic to nonisotropic flow, depends on N. For small val-

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FIG. 1. ͑Color͒ Pressure and magnetic induction fields for selected cases in Table I: ͑a͒ f1-0-250 at tϭ0, ͑b͒ f1-0-250 at tϭ15, ͑c͒ f1-45-250 at tϭ0, ͑d͒ f1-45-250 at tϭ15, ͑e͒ f3-0-250 at tϭ0, ͑f͒ f3-0-250 at tϭ15, ͑g͒ f3-45-250 at tϭ0, ͑h͒ f3-45-250 at tϭ15. Note that the time t is in units of the eddy turnover time. ues of N, the flow remains essentially isotropic. Intermediate elongated in this direction. The present effort builds on this N values lead to organized quasi-two-dimensional evolution analysis by considering compressibility and lifting the dual lasting several eddy turnover times, which is interrupted by constraints of quasistatic simplification and the use of forc- strong 3D turbulent bursts. For large values of N, there is ing to maintain statistical steady state. strong anisotropy and significant suppression of turbulence, The inverse cascade of magnetic helicity has also re- 23 with the flow restricted to the perpendicular direction to B0 . ceived some attention. The significance of this phenom- That is, only velocity modes with nonzero gradients in the enon lies in the provision of a mechanism for generating ͑ ͒ direction of B0 is Joule dissipated; the velocity gradients in large-scale fluctuating magnetic energy. The phenomenon → ϱ͑ the direction of B0 are damped and vortical structures are has been reported for the case where B0 0at nonhomo-

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FIG. 2. Evolution of the kinetic energy Ek . Refer to the text and Table I for the various cases deﬁned by the legend.

͒ ϭ ͑ ͒ 24 TABLE I. Case studies for the MHD turbulence DNS calculations. geneous or B0 0 homogeneous . A proposed mechanism starts out with the transfer of large scale momentum energy ␪ Case No. Case Reference f u , f b °Re␴ Remarks to the small scale momentum field. This is followed by a I f1-45-250 2, 2 45 250 transfer of the energy to the small-scale magnetic field via II f1-45-1 2, 2 45 1 the equipartition action of the Alfveń waves at the high wave III f1-45-100 2, 2 45 100 numbers. The final step in the process is the back transfer of IV f1-0-1 2, 2 0 1 V f1-0-100 2, 2 0 100 the small-scale magnetic energy to the large-scale magnetic VI f1-0-250 2, 2 0 250 energy field. VII f3-45-1 3, 1 45 1 The three scaling laws of Kolmogorov for non-MHD VIII f3-45-100 3, 1 45 100 turbulence have also been investigated for MHD turbulence. IX f3-45-250 3, 1 45 250 Ϫ X f3-0-1 3, 1 0 1 An energy spectrum with an exponent of 2/3 has been 25 XI f3-0-100 3, 1 0 100 reported, in contrast to the Ϫ5/3 law. This result is in XII f3-0-250 3, 1 0 250 agreement with those of Kraichnan26 and Iroshnikov27 MHD XIIIa f1-45 2, 2 45 N/A No MHD ͑ a results for the second law of Kolmogorov temporal decay of XIV f1-0 2, 2 0 N/A No MHD ͒ XVa f3-45 3, 1 45 N/A No MHD energy have been obtained using the idea of the persistence XVIa f3-0 3, 1 0 N/A No MHD of large scales together with results from Kraichnan and Ironshnikov. Compared to the non-MHD case, this leads to a a Cases XI–XIV do not involve MHD, although the initial conditions were different expression for the temporal decay. Gomez, calculated to match those for the corresponding MHD cases. Note that Re␴ 25 is not used in the calculation of the initial conditions and that the param- Politano, and Pouquet validated the foregoing results, as ␪ ͑ eters f u , f b , , and Re␴ have been introduced earlier in this paper. well as the MHD results for the structure functions third

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2 FIG. 3. Evolution of the square of the sonic Mach number, M s . Refer to the text and Table I for the various cases deﬁned by the legend.

law͒ in terms of the Elsa¨sser variables zϮϵuϮb. However, implication that turbulence-Alfveń wave interaction does not more recent results by Muller and Biskamp28 seem to sug- control the turbulence dynamics, contrary to earlier sugges- gest that MHD turbulence with finite magnetic helicity does tions. not follow the phenomenology of Ironshnikov or Kraichnan, Concerning intermittency, both temporal and spatial in- but agrees more with the Kolmogorov scaling. There is the termittency have been reported for turbulent MHD flows,29

TABLE II. Global statistics of the initial conditions. The initial conditions for cases XI–XIV match those for the corresponding MHD cases and are not repeated here.

͗ " ͘ϵ ͗␳ " ͘ 2 ͗ " ͘ Case No. M s u b 2Hc u u /2 ͗u ͘ b b /2 Ei Rub I 0.233 0.25 0.1551 0.25 0.125 0.030618 1.00 II 0.233 0.25 0.1551 0.25 0.125 0.030618 1.00 III 0.233 0.25 0.1551 0.25 0.125 0.030618 1.00 IV 0.224 2.979ϫ10Ϫ4 0.1762 0.25 0.125 0.078395 1.193ϫ10Ϫ3 V 0.224 2.979ϫ10Ϫ4 0.1762 0.25 0.125 0.078395 1.193ϫ10Ϫ3 VI 0.224 2.979ϫ10Ϫ4 0.1762 0.25 0.125 0.078395 1.193ϫ10Ϫ3 VII 0.433 0.333 0.7782 1.00 0.05556 0.06209 1.00 VIII 0.433 0.333 0.7782 1.00 0.05556 0.06209 1.00 IX 0.433 0.333 0.7782 1.00 0.05556 0.06209 1.00 X 0.408 3.973ϫ10Ϫ4 0.9283 1.00 0.05556 0.08563 9.27ϫ10Ϫ4 XI 0.408 3.973ϫ10Ϫ4 0.9283 1.00 0.05556 0.08563 9.27ϫ10Ϫ4 XI 0.408 3.973ϫ10Ϫ4 0.9283 1.00 0.05556 0.08563 9.27ϫ10Ϫ4

FIG. 4. Evolution of the turbulence Reynolds number normalized by the initial values of the root mean-squared velocity. Refer to the text and Table I forthe various cases defined by the legend. with 3D flows less intermittent than 2D, but more intermit- jets and the bifurcation of eddies in the compressible flow tent than 3D non-MHD turbulence. The physical variables field. The studies were extended to the supersonic regime in (u,b,zϮ) tend to be more intermittent in the MHD case than a follow-up paper.31 in non-MHD and magnetic field could be as intermittent as Scaled and spectrally-filtered random initial conditions the vorticity field. for the velocity and magnetic fields appear to be more suit- The foregoing results pertain mostly to incompressible able for aerospace applications33 than the nonrandom flows and therefore are not directly useful for aerospace ap- Orszag–Tang vortex.29 The systematic compressibility stud- plications, where the flow is compressible. Dahlburg and ies by Ghosh and Matthaeus32 ͑hereafter referred to as GM1͒ Picone,30 Picone and Dahlburg,31 and Ghosh and used the former type of initial conditions and considered the Matthaeus32 have investigated compressible MHD flows, but characterization of three distinct time asymptotic types of they focused on nonaerospace applications. Dahlburg and behavior. Results for initial Mach numbers of 0.1 and 0.5 Picone30 examined the evolution of the Orszag–Tang were compared to the incompressible results for a fixed value 29 ϭ vortex in a 2D compressible medium. The initial conditions of the Alfveńic Mach number M 2 1.0 and magnetic for the system consist of a nonrandom periodic field in which Reynolds number Re␴ϭ250. Three regions in the E/A Ϫ 4 ͑ the magnetic and velocity fields contain X points but differ in E/2Hc diagram of Ting et al. were examined. E is the modal structure along one spatial direction. Calculations total energy, A is the mean squared magnetic potential, and ͒ were done for viscous and resistive Lundquist numbers of Hc is cross helicity. The study reported selective decay of 50, 100, and 200, with Mach numbers in the range 0.1 kinetic energy when kinetic and magnetic energies were ini- р р M a 0.6. Compressible effects, which develop within one tially approximately equipartitioned. When the intensity of or two Alfveń transit times in this system, include the retar- the magnetic fluctuations (brms) was significantly smaller dation of the growth of the correlation between u and b, the than that of the velocity fluctuations (urms), a fast temporal emergence of compressible small-scale structure as massive dissipation of the magnetic energy was observed. The third

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FIG. 5. Evolution of the longitudinal component of the velocity, UL. Refer to the text and Table I for the various cases defined by the legend. region examined in GM1 is characterized by highly corre- current limitations in computer speed. Despite access to the lated u and b fields in the initial flow. In time, the two fields most powerful supercomputers and a high-order accurate tend to reach a constant energy ratio ͑Alfveń ratio͒ of order fully 3D simulation tool,43 it was found unfeasible to explore unity. The spectra of the kinetic, magnetic, and internal en- the range of parameters of interest. Recent efforts using mul- ergies were reported for the three regions and were used to tiprocessing strategies44 show promise of lifting these con- demonstrate the appearance of acoustic turbulence in the first straints and we expect to obtain selective 3D results in the and second regions discussed above. future. Note that studies based on the quasistatic approxima- The present paper differs from foregoing literature in tions, such as in Kassinos, Knaepen, and Carati,34 can afford several aspects. First, the emphasis on aerospace applications to calculate full 3D models because the MHD transport equa- is new and implies small relative values of the electrical tions are not solved. Note that the simulations in GM1 as conductivity ␴, as measured with the magnetic Reynolds well as those in the papers by Dhalburg and Picone, included number. For example, it is six to seven orders of magnitude the MHD equations and were also based on a 2D model. smaller than in the astrophysical applications examined in Nevertheless, they reported reasonable results that agree well GM1. It is also of interest to investigate the sensitivity of the with the expected physics, with the suggestion that a care- results to a range of Re␴ values within 1.0рRe␴р250.0. ͑In- fully studied 2D model could provide useful information on clusion of Re␴ values in the high end of this range allows the real system. In order to reduce spectral energy back comparison with the results in GM1.͒ transfer resulting from the use of 2D calculations, we remove An obvious limitation of the present work is the use of a the first spectral mode when calculating some of the turbu- 2D model, which requires that care be exercised in interpret- lence quantities. ing the results. This is a compromise, however. The number The governing equations are presented in Sec. II, fol- of parameters of dynamic significance is numerous, which, lowed by a detailed description of the initial condition speci- together with the inclusion of the MHD transport equations, fication and generation in Sec. III. Section IV summarizes prevents any extensive parametric 3D studies because of the the numerical procedure for calculating the various equa-

FIG. 6. Evolution of the transverse component of the velocity, UT. Refer to the text and Table I for the various cases deﬁned by the legend.

␳E 1 1ץ -tions, while Sec. V discusses the various case studies. Re ϩٌ ͑␳EϩP͒uϪR ٌ Bͩ u Bͪ Ϫ ٌ ͑u"␶͒ • t • b • • ␮ Reץ -sults are presented in Sec. VI, followed by concluding re marks in Sec. VII. m 1 Ϫ ٌ k T ͑␥Ϫ1͒Pr M 2 Re • “ II. GOVERNING EQUATIONS 1 R 1 1 1 1 ͒ ͑ The MHD equations are solved in dimensionless form, ϩ b ٌ ͫ " Ϫ ͬϭ • ␮ ␴ B “ ␮ B “ ␮ B• ␮ ␴ B 0. 4 ␳ ␮ Re␴ m m m m with the scales L0 , U0 , L0 /U0 , 0 , P0 , T0 , B0 , k0 , 0 , ␮ , and ␴ for length, velocity, time, density, pressure, m0 0 temperature, magnetic induction field, thermal conductivity, The governing equations for the magnetic field induction dynamic viscosity, magnetic permeability, and electric con- have been obtained by combining the Ampe`re–Maxwell ductivity, respectively. The equations take the following Law, Faraday’s Law, and Ohm’s Law. Although the total form: energy equation is solved in the form of Eq. ͑4͒ above in our ␳ code, the nondimensional equation of state used in theץ ␥ ϩٌ ͑␳u͒ϭ0, ͑1͒ t • present paper is pϭ␳ , meaning that Eq. ͑4͒ is internallyץ transformed in our code to be consistent with the equation of ␥ R 1 ץ ␳ ͒ϩٌ ␳ Ϫ b ϩ Ϫ ٌ ␶ϭ ͑ ͒ state. , the ratio of specific heats, is kept constant at 5/3 for͑ u ͩ uu ␮ BB PIͪ 0, 2 ץ t • m Re • all the results in this paper. The length scale is the integral ␲ 2 ͐ϱ␬Ϫ1 ␬ ␬ scale of the initial field, or /2urms 0 E( )d , where B 1 1 B ␬ץ -ϩٌ ͑ Ϫ ͒ϩ ٌϫ ٌϫ ϭ ͑ ͒ urms and E( ) are the respective values of the root mean uB Bu ͫ␴ ͩ ␮ ͪͬ 0, 3 ץ t • Re␴ m squared velocity fluctuation and energy spectrum of the ini-

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FIG. 7. Evolution of the internal energy Ei . Refer to the text and Table I for the various cases deﬁned by the legend.

tial field. Note that the length scale has a nondimensional ␶ϭ␮͑ ϩ *͒Ϫ 2␮ٌ “u “u 3 •u, value of unity at all times, even though the integral length is time-dependent. where an asterisk on a quantity denotes a transpose operation The independent variables of the problem are time, t, on the quantity. Other parameters can be defined, such as the ϵ ͑ and the spatial coordinate directions x (x,y,z). The pri- characteristic Alfveńic Mach number M 2 Ghosh and ␳ 32͒ ϭ ␮ Ϫ1/2 mary dependent variables are density , the velocity compo- Matthaeus , which can written as M 2 (Rb / m) . Also, ϵ ϭ ␳ ␮ 1/2 nents u (u,v,w), the components of the magnetic induc- the use of the Alfveń’s velocity UA B0 /( 0 m ) in the tion field, Bϵ(B ,B ,B ), and the total energy, ␳E. The 0 x y z definitions of the hydrodynamic ͑Re͒ and magnetic (Re␴) temperature, T, also appears, as does the reduced pressure Reynolds numbers, respectively, gives the viscous and resis- ϭ ϩ ͉ ͉2 ␮ ␳ ϭ͓ ␥Ϫ ͔ P p Rb( B /2 m) and total energy E p/( 1) tive Lundquist numbers S and S used by Dahlburg and ϩ 1␳͉ ͉2ϩ ͉ ͉2 ␮ v r 2 u Rb( B /2 m). The parameters of the problem in- 30 ϭ Picone. This equivalence occurs when M 2 1. It is easily clude the nondimensional thermal conductivity, k, dynamic shown that the instantaneous value of the interaction param- ␮ ␮ viscosity, , magnetic permeability, m , electric conductiv- eter, N,isN ␴/(u ␳), where N is the product of the mag- ␴ ϭ ␮ 0 rms 0 ity, , the Prandtl number, Pr C p 0 /k0 , where C p is the netic Reynolds number and the magnetic pressure number in specific heat at constant pressure, the characteristic sonic the initial field. Thus, with R ϭ1 in the initial field, 1рN ϭ b 0 Mach number, M 1 U0 /Cs , where Cs is a characteristic р 250. The nondimensional eddy turnover time, TE , is equal sound speed, the magnetic pressure number, Rb to 1/͗urms͘, where urms is the nondimensional root-mean ϭB2/␳ U2␮ , the hydrodynamic Reynolds number, Re 0 0 0 m0 squared turbulence velocity. The nondimensional calculation ϭ␳ ␮ 0U0L0 / 0 , and the magnetic Reynolds number, Re␴ time, that is, the time that appears in the governing Eqs. ϭU L /(␮ ␴ )Ϫ1ϵU L /␩ . The viscous stress tensor ␶ has ͑ ͒ ͑ ͒ 0 0 m0 0 0 0 0 1 – 4 , is denoted by t. The number of eddy turnover time its usual definition: units is therefore equal to t͗urms͘, where

ϵ " ␮ FIG. 8. Evolution of Eb Rb͗b b͘/2 m .

1 t energy in each mode. This ‘‘tophat’’ spectra initial energy ͗ ͑͘ ͒ϭ ͵ ͑ ͒ urms t urms t dt. distribution has been widely used in compressible turbulence t 0 literature,32,35,45 as well as in DNS of chemically reacting Note that this integral gives the ͑nondimensional͒ distance flows. The spectrum, which is basically a filter for the ‘‘en- travelled by an eddy over a ͑nondimensional͒ time period t. ergy’’ from the random fields, allows only a few modes in Standard averaging over the grid points is used to compute the initial field ͑approximately 10 in 2D or 3 in each of the urms at each time step and urms is used to represent the ve- two coordinate directions͒. A detailed description of the gen- 35 locity of an eddy at each instant of time. Hence, ͗urms͘(t) eration of this aspect of the initial conditions is available. represents a pseudoconvective velocity at time t. Information The correlation between the otherwise random velocity on the variation of urms with time can be gleaned from the Re and magnetic induction fields has been obtained using two ϭ plot since Re(t) Re0*urms(t) /urms(tϭ0) , where Re0 denotes different methods. The first consists of generating the ran- the value of the specific ͑initial͒ hydrodynamic Reynolds dom fields in Fourier space and introducing a phase angle number. between the two fields ͑GM1͒. Random numbers for vorticity ␻͑k͒ and the magnetic potential a(k) are generated: III. INITIAL CONDITIONS ␻͑ ͒ϭ ͉ ͉ i͓2␲R͑k͔͒Ϯ␪ The calculation of the initial fields u and b starts with k C1 ͚ k e , 1р͉k͉2р12 the generation of random numbers in ͑Ϫ1,1͒ for the fields. The corresponding spectral fields uˆ and bˆ are generated and 1 ͑ ͒ϭ i͓2␲R͑k͔͒ the incompressible components uˆ IϭuˆϪ(k"uˆ/k2)k, bˆ Iϭbˆ a k C2 ͚ ͉ ͉ e , 1р͉k͉2р12 k Ϫ(k"bˆ/k2)k are determined, where k is the wave number vector, with magnitude k. These fields are then filtered with where R(k) is a unique random number for each wave num- the energy Eϭ1 for 1рkрͱ12 and then scaled with the ber vector k, and ␪ is the phase angle between the initial

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FIG. 9. Fractional contribution of the turbulence kinetic energy to the total energy. Refer to the text and Table I for the various cases defined by the legend. vector potential and vorticity. The corresponding velocity aware of these correlations in the literature. Note that the ͗ " ͘ϭ͗ " ͘ϭ 2ϭ 2ϭ and magnetic induction fields are obtained as conditions u u b b 1 can be obtained with f u f b rA ϭ1. The cases ␪ϭ0° and ␪ϭ45° are studied in this paper. It u͑k͒ϭkϫzˆ␻͑k͒/͉k͉2, is pointed out that ␪ϭ0° results in an identity transformation b͑k͒ϭkϫzâ͑k͒/͉k͉2. in Eq. ͑5͒, meaning that the original randomly generated u and b fields are used in this particular case. An alternative approach involves taking the physical space The velocity and magnetic induction fields obtained random results uЈ and bЈ and correlating them: from the foregoing procedure are used in the calculation of u cos ␪ sin ␪ uЈ the initial fluctuating pressure field p(x) via the incompress- ͫ ͬϭͫ ͬͫ ͬ. ͑5͒ b sin ␪ cos ␪ bЈ ible flow relation

The results reported in this paper were obtained with this Rb Rb ␳uu͒ϩٌ2pϪ ٌ ͑bb͒ϩ ٌ2b2ϭ0. ͑6͒ ٌ͑ ٌ approach. The correlated fields are subsequently scaled: • • ␮ • 2␮ m m u u/(f uurms) and b b/(f bbrms), where urms ϭ ͉ ͉2 1/2 ϭ ͉ ͉2 1/2 ͓͗ u(x) ͔͘ and brms ͓͗ b(x) ͔͘ . For the procedure The pseudosound density correction is obtained using the ϵ 35 outlined above, we have found that the quantities f f b f u procedure for the non-MHD SVPS case in Ladeinde et al. and ␪ are very important to the statistics of the initial field, For decaying turbulence simulation such as the present ϭ " " such as the Alfveń ratio rA ͗u u͘/͗b b͘, cross helicity Hc one, the state of the initial flow determines the evolution ϭ͗u"b͘/2, the rms values of the flow and magnetic induction pattern for prescribed parameter values. In the present study, fields, and the initial eddy turnover time scale, TE . Specifi- a wide range of statistics has been investigated. The total " ϭ ␪ϭ ϭ ϩ ϩ cally, the simple relation, ͗u b͘ 1/f results for 45°, energy Et is defined as Et Ek Eb Ei , where Eb " ϭ " ϭͱ ϭ 2 ͑ ϭ ␮ " while ͗u b͘30° ͗u b͘60° 3/2f and rA ( f b / f u) . Here, (Rb / m)͗b b͘/2 is the magnetic energy and Ei is the in- ͗u"b͘␾ denotes the value of ͗u"b͘ when ␪ϭ␾°.͒ We are not ternal energy associated with the polytropic equation of state:

FIG. 10. Fractional contribution of the ﬂuctuating magnetic energy to the total energy. Refer to the text and Table I for the various cases deﬁned by the legend.

␥ ␥ ͑͗␳ ͘Ϫ͗␳͘ ͒ ␳␥ 2 ϵ ͑ ͒ 1 Ei , 7 ͑ ͒ϭ ͚ ͯ ͩͱ ͪͯ 2 Pi i FFT , M ␥͑␥Ϫ1͒ n Ϫ р р ϩ 2␥͑␥Ϫ ͒ 1 i 1/2 k i 1/2 M 1 1 where n refers to the number of modes in an annulus for the bin in spectral space. The kinetic and magnetic energy spec- where ␥, the ratio of specific heats, is specified as 5/3 every- tra P (i) and P (i) were also calculated: where in the calculation domain. The instantaneous sonic k m number of the turbulent flow is defined as M s(t) ␥Ϫ ϭu (t)/͗c (x,t)͘, with c (x)ϭ␳(x)( 1)/2/M . Note that 1 rms s s 1 ͑ ͒ϭ ͉ ͑ͱ␳ ͉͒2 in simulations where ͗u"u͘ϭ1 for the initial field and M Pk i ͚ FFT u , s n iϪ1/2рkрiϩ1/2 ϭ͗ 2 ͘1/2 Þ M s (x) , M 1 M s because the pseudosound density correction is nonzero. Therefore, the deviation of M s from M 1 is a statement of the integral deviation of the local den- 1 ͒ϭ ͉ ͉͒2 ␳ϭ Pm͑i ͚ FFT͑b . sity from 1, or the pseudosound density correction from n Ϫ р р ϩ ϭ i 1/2 k i 1/2 zero. On the other hand, the condition M 2 M a will be satisfied for the simulations in which ͗b"b͘ϭ1inthe ϭ͗ 2 ͘1/2 initial field, where M a M a(x) and M a(x) Other quantities that have been studied in this paper as a ϭ ␳1/2 ͉ ͉ ͉ ͉ L M 2 (x) u(x) / b(x) . function of Re␴ include the longitudinal U and transverse The spectral internal energy distributions have been ob- UT velocity components. tained in this work as a way of assessing the level of acoustic An attempt was made to locate the parameter space for turbulence. This quantity is obtained as our calculations inside Fig. 1 of GM1, which was originally

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FIG. 11. Fractional contribution of the internal energy to the total energy. Refer to the text and Table I for the various cases defined by the legend. developed for incompressible MHD flows. Because b and u hand. This choice has been motivated by the need to extend in the initial fields are incompressible in the present work, it our analysis to engineering problems with more complex is the initial fields for ␳ and p that determine whether we stay geometrics and boundary conditions and for which the spec- within the diagram. We have found that many ␳ fields satis- tral method does not perform efficiently. The implementation fying the pseudosound initial conditions used in this paper of the high-order Pade´ schemes in Gaitonde and Visbal36 is ͑ ͒ ͑ ͒ give values of E/A and E/2Hc for the resulting compressible used, whereby Eqs. 1 – 4 are written in the following field ͑i.e., in terms of the compressible analogs of these two strong conservation form in generalized curvilinear coordi- variables͒ that are located outside of the diagram. No attempt nates ͑␰,␩,␨͒: was made in this paper to force the compressible data to fit into the chart. The initial fields for pressure and the magnetic HЈץ GЈץ FЈץ HЈץ GЈץ FЈץ XЈץ induction are shown in Fig. 1, along with the distribution of ϩ ϩ ϩ ϭ v ϩ v ϩ v ϩ Ј , ␨ Sץ ␩ץ ␰ץ ␨ץ ␩ץ ␰ץ ␶ץ the fields when tϭ15. Tables I and II show some global statistics of the initial fields. ͑8͒ Finally, removing the nonsolenoidal component of b -ϳ Ϫ6 where ␶ is the time domain, ͑␰,␩,␨͒ are the transformed co ٌ produces a field with •b 10 compared to unity, whereas -ϳ ordinates that respectively correspond to the physical coordi ٌ •b O(1), otherwise. nates (x,y,z). FЈ, GЈ, and HЈ are vectors of the inviscid fluxes while FЈ , GЈ , and HЈ are the fluxes associated with IV. NUMERICAL SOLUTION v v v the stresses, and SЈ represents the source terms in the equa- The majority of fundamental simulation of MHD tions. Our code also allows the use of the weighted essen- turbulence29,30,32 have been based on the spectral method. tially nonoscillatory ͑WENO͒ finite difference schemes37 The present investigation, however, is based on the finite- when the Mach number is high enough to introduce shock difference method, which is less efficient for the problem at waves into the calculations. Although the results in the

FIG. 12. Evolution of the total energy. Refer to the text and Table I for the various cases defined by the legend. present paper are for regular geometries and moderate Mach With the compact schemes, the derivative uЈ for a ge- numbers, the same formulations have been used for DNS in neric variable u is obtained from the representation non-Cartesian geometries38 and flow fields with shock 39–41 Ϫ Ϫ waves. uiϩ2 uiϪ2 uiϩ1 uiϪ1 ␣uЈ ϩuЈϩ␣uЈ ϭb ϩa , iϪ1 i iϩ1 4⌬␰ 2⌬␰ ͑9͒ TABLE III. Partition of the total energy into its components at the end of simulation. Note that Ei is oscillatory, so that the data in this table cannot be where ␣, a, and b are constants that determine the spatial used to predict a trend involving Ei . The figures for the evolution of this properties of the algorithm. The base compact differencing quantity should be used in conjunction with this table. schemes used in this paper are the three-point, fourth-order ␣ ϭ 1 3 Case reference Ek(%) Eb(%) Ei(%) Et scheme, C4, with ( ,a,b) ( 4, 2,0), and the five-point, ␣ ϭ 1 14 1 f1-45-1 96.59 0.862 3.323 0.0859 sixth-order scheme, C6, with ( ,a,b) ( 3, 9 , 9). Note that f1-45-100 56.65 25.854 17.50 0.0606 the symbol u above also represents components of vector f1-45-250 51.814 44.44 3.75 0.0777 quantities, such as the FЈ vector. f1-0-1 79.88 0.601 20.00 0.12835 The formula in Eq. ͑9͒ is used to calculate the various f1-0-100 28.54 45.69 25.77 0.03398 ͑␰ ␩ ␨͒ f1-0-250 25.90 58.15 15.94 0.05259 derivatives in the , , plane, as well as the metrics of the f3-45-1 97.39 .0755 2.60 0.4617 coordinate transformation. The derivatives of the inviscid f3-45-100 82.475 0.9 16.62 0.2406 fluxes are obtained by first forming these fluxes at the nodes f3-45-250 70.63 18.18 11.18 0.1149 and subsequently differentiating each component with the f3-0-1 91.61 0.0647 8.39 0.6168 above formulas. In order to reduce error on stretched meshes, f3-0-100 87.03 2.675 10.29 0.2505 f3-0-250 75.30 13.703 10.99 0.0896 the required metrics are computed with the same scheme that is employed for the fluxes. To form the second derivative

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FIG. 13. Comparison of the fractional contribution of the three energy components for each MHD case study. Refer to the text and Table I for the various cases defined by the legend. terms, the primitive variables are first differentiated and ployed to numerically stabilize the compact differencing cal- properly combined with the transport coefficients to form the culations, as discussed in detail by Gaitonde and Visbal.36 requisite combinations of first derivative terms. These gradi- Time advancement is based on the fourth-order, four-stage, ents are then differentiated again with the same difference Runge–Kutta method. scheme. The physical boundary conditions are applied after The WENO schemes37 pertain to the differencing of the each update of the interior solution vector. Filters are em- convective terms of the equations in a way that avoids dis-

FIG. 13. ͑Continued.͒

cretization across strong gradients. ͑The viscous terms are scales which, for the present work, could allow up to four discretized with fourth order central schemes.͒ We can write energetic eddies in a direction of the computational box. The the WENO problem for Eqs. ͑1͒–͑4͒ as largest grid size that can be used to resolve the smallest scale of flow was calculated using standard procedures, such as on u ͑␰,␩,␨,t͒ϩ f ␰͑u͑␰,␩,␨,t͒͒ϩg␩͑u͑␰,␩,␨,t͒͒ t page 347 of the text by Pope.42 According to this calculation, ϩh␨͑u͑␰,␩,␨,t͒͒ϭ0, 128 spectral modes are sufficient in each direction. Prelimi- 2 2 2 where u, f, g, and h are vector functions. The convective nary calculations with resolutions of 512, 256, and 128 terms are reconstructed to accuracy k, which is taken as 6 in finite difference cells show that the scales of flow were re- the present work. The basic WENO differencing problem is solved down to the Kolmogorov scale by all resolutions. Most of the calculations reported in this paper were carried du 1 1 2 ijk ϭϪ ˆ Ϫˆ ͒Ϫ out using 256. However, postprocessing the data for the ͑ f iϩ͑1/2͒, j,k f iϪ͑1/2͒, j,k ͑gˆ i, jϩ͑1/2͒,k dt ⌬␰ ⌬␩ plots was based on a 642 grid in order to reduce the compu- 1 tational cost which otherwise tends to be greater than that for Ϫ ˆ ͒Ϫ ͑ ˆ Ϫ ˆ ͒ gi, jϪ͑1/2͒,k ⌬␨ hi, j,kϩ͑1/2͒ hi, j,kϪ͑1/2͒ the basic calculations. Using the reduced data sets in this manner did not introduce significant errors. Figures 1͑a͒– ϭ f ЈϩgЈϩhЈϩO͑⌬␰k,⌬␩k,⌬␨k͒, 1͑h͒ show the initial magnetic field arrow plots and the pressure contour maps for the cases f1-0-250, f1-45-250, f3-0- ϵ ͑ ͒ L u , 250, and f3-45-250 at tϭ0 and midway through the where ˆf , gˆ , and hˆ are high-order representations of the in- calculations, at tϭ15. Note that the magnetic field, as well as viscid fluxes that are generated by the use of Lagrange inter- the velocity field ͑not shown͒ show approximately between 2 polation. The reader is referred to Ref. 39 for the details of and 4 eddies in each direction of the box. The pressure con- our implementation. The computational domain is a 2␲ϫ2␲ tours show a much larger dimension for the structure because periodic square. The reference length scale is the integral the total field (͗p͘ϩpЈ) is shown. The displayed pressure

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FIG. 14. Energy spectra at tϭ5, 15, and 30 for f3-45-1.

field is dominated by the underlying mean component which, studies is bigger or at least comparable to those that have depending on the particular initial condition ͓such as that in been used in many of the MHD studies we reviewed.30–32,34 Fig. 1͑c͔͒, lacks a spectral distribution. More importantly, for the target application ͑hypersonic flight͒, it is generally supposed that the turbulence intensity V. CASE STUDIES is low, so that the value of the turbulence Reynolds number used in the present study might be relevant to the real sys- The large number of parameters controlling the dynamics of the problem under investigation is evident. The calcu- tem. ϭ Sixteen cases were investigated, as summarized in lations are limited to M a 0.5. This choice is guided by the results in GM1 and from our own work,35,40 in which the Tables I and II. They are referenced with two or three codes р р that are separated by hyphens; for example, f3-45-250. The lower M a calculations (0.1 M a 0.3) gave results that were similar to those of the corresponding incompressible first code roughly describes the relative intensity of the fluc- cases, and higher M runs produced shocklets and other po- tuating velocity and magnetic induction fields, where f1 im- a ϭ ϭ tentially discontinuous fields that require a different kind of plies f b / f u 1 and f3 implies f b / f u 3. Note that f u and f b analysis from the one presented in this paper. The effects of control the intensity of turbulence in the initial fields for u the initial hydrodynamic turbulent Reynolds number is also and b. The second code in this example, 45, denotes the well known from previous work. This parameter is also kept value of the angle ␪ used in the correlation of the initial u ␮ ␮ ␴ fixed at a value of 250. The parameters k, , m , and are and b fields, while the third code, for example 250, denotes set to unity, implying constant properties. The parameters the value of the magnetic Reynolds number, Re␴ . Cases ␪ varied in this study are, therefore, f u , f b , , and Re␴ .Itis XIII–XVII are non-MHD that have been included to assess pointed out that the initial hydrodynamic Reynolds number, the effects of MHD on the various results. These cases do not Re, is based on the root-mean squared turbulence velocity, involve Re␴ . We investigated two values of ␪: 0° and 45°. ␪ urms , and a length scale of unity relative to the integral These lead to interesting initial correlations. Note that val- length scale. The value of 250 that is used in the present ues of 30°, 60°, 120°, etc., give initial fields that are not

FIG. 15. Energy spectra at tϭ5, 15, and 30 for f3-45-100.

͑ ͒ ͑ ͒ dynamically different. The choice of the values for f u and f b Figures 2 a –2 d show the temporal evolution of the ϵ ␳ " was based on a preliminary work that we carried out which kinetic energy, Ek(t) ͗ u u͘/2 from the respective values ϭ ͑ showed that a field is complicated to calculate when brms is of Ek(0) 0.155, 0.176, 0.778, and 0.928. The f1 case for large. We also benefitted from discussions with Dr. S. Ghosh both ␪ϭ0° and ␪ϭ45°͒ show identical evolution for Re␴ϭ1 on this matter. Concerning the Re␴ values, we wanted a and non-MHD flows, Figs. 2͑a͒ and 2͑b͒, whereas, for f3, the range that included low values, which are more relevant to non-MHD runs have a visibly faster decay rate. Note the aerospace applications, and values that are comparable to difference in the scales of the graphs on the y-axis. This those that have been studied in previous work, to provide implies that the above difference in decay rates is significant some basis for evaluating our calculations. and suggests that the effects of magnetic field on the evolu- VI. RESULTS AND DISCUSSIONS tion of the turbulence energy is negligible at the lowest magnetic Reynolds number investigated. The results also suggest The computer program used for the present investiga- that at the higher Re␴ values, when magnetic effects are im- tions has been thoroughly validated in order to engender con- 43,44 portant, the magnetic field tends to accelerate the decay of fidence in the results it produces. The basic compressible the turbulence energy relative to non-MHD flows. The nature turbulence procedure, as well as the procedure to extract tur- of this enhancement in decay rate depends on the particular bulence statistics have been validated by comparison of the Re␴ values, the relative initial intensity of the u and b fields, results for various methods ͑ENO, WENO, compact schemes͒ to the pseudospectral calculations in GM1.35,39,45 and the correlation between the fields. For f3, the decay rate For the MHD simulation, we have compared predicted re- increases very rapidly with increasing Re␴ value, whereas sults using the program with analytical solutions for un- the f1 cases show a nonmonotonic trend, with Re␴ϭ100 de- steady Alfveń waves with Ohmic damping.41 A few of the caying faster than Re␴ϭ250. The reason for this peculiar results in this paper ͑e.g., the kinetic energy spectra͒ also behavior is not apparent, but we will return to this point later agrees with those that use similar conditions in GM1. in the paper.

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FIG. 16. Energy spectra at tϭ5, 15, and 30 for f3-45-250.

ϭ The foregoing trends for Ek(t) are mirrored in the evo- 0.05. The internal energy directly measures compressibility 2 ͑ ͒ lution of M s Fig. 3 , the instantaneous turbulence Reynolds effects. Hence, the striking similarity in the evolution pattern ͑ ͒ L ͑ ͒ number normalized by the initial urms value, Re, Fig. 4 ,as for U and Ei(t) Fig. 7 . Because of the relatively low well as the transverse component of the velocity field, UT, Mach number, density fluctuations are small compared to the which is calculated as 2ϳ 2 velocity fluctuations. Therefore, M s urms. Also, Ek(t) T Ϫ1 I 2 1/2 ϳ 2 ϳ U ϭ͕͉͗FFT ͑uˆ ͉͒ ͖͘ . urms and Re urms by their definitions. These scalings ex- 2 L plain the similar trends observed for M s , Ek(t), and Re. The The evolution of the longitudinal component, U , obtained evolution of UT ͑Fig. 6͒ is more difficult to explain. How- by substituting uˆ C for uˆ I in this equation, is completely dif- ever, because the Mach number is not large, the incompress- ferent from that for E (t). Note that uˆ C and uˆ I are the spec- k ible component dominates the large scale velocity field; tral compressible and incompressible components of the ve- hence UT scales with u and the evolutionary trend for UT locity field, respectively. UL shows an oscillatory pattern rms should be similar to that for the other variables just dis- ͑Fig. 5͒, with magnitudes which are two to four times smaller than those for UT ͑Fig. 6͒. This oscillation is ex- cussed. On the other hand, Ei(t) scales with the density fluc- pected if one considers that the compressible component tuation, and is therefore a direct measure of the extent of usually propagates in waves.35 The results for non-MHD and compressibility in the flow. Thus, the similar evolutionary L for Re␴ϭ1 are relatively close, as has been observed above trends for U and this quantity. Concerning Re evolution, the for other variables. With the exception of the f1-45 cases, f1 cases appear to start from a value of 125 in the plots ͑Fig. which show continuous decay, the average ͑i.e., smoothened͒ 4͒, as opposed to the value of 250 that was imposed. For profile ͑not explicitly shown͒ is characterized by an initial these cases, the specified Reynolds number, Re, is in fact elevation with time, followed by a continuous decay. The equal to 250. However, the short time value of urms is ap- profiles in Figs. 5͑b͒–5͑d͒ appear to approach an equilibrium proximately equal to 0.5, so that the effective ͑short time͒ value of ULϭ0.1, whereas the f1-45 cases approach UL Reynolds number is 125.0, which is what shows up in the

FIG. 17. Energy spectra at tϭ5 and 30 for f1-0-100 and f1-45-100. plots. Because of this, some care is required when interpret- The initial jump occurs earlier than tϭ1. The f3 calculations ing the results, although the general conclusions from the for both ␪ values also show a similar trend, with the jump present studies remain unchanged. To have an effective occurring at a slightly later time (tϭ4 for f3-45-250 and t Reynolds number of 250 at short time, the imposed value ϭ2 for f1-0-250͒. An important conclusion from Fig. 8 is will have to be 500, which would still conflict with the value that higher Re␴ values significantly slow down the magnetic ͓ ϭ for the f3 cases, which is 250. Note that (urms)t→0 1.0 for energy decay rates, thereby prolonging the time it takes for the f3 cases.͔ Thus, the instant dip in the effective Re value the magnetic energy to die off completely. At the lowest Re␴ for the f1 cases seems to be an unavoidable feature of the value investigated, the magnetic energy decays to zero al- parameter space investigated. most instantly. Note also that the Eb decay mode is less The evolution pattern for the magnetic field energy, ␪ ϵ " ␮ sensitive to the initial correlation angle , in comparison to Eb(t) Rb͗b b͘/2 m , is quite different from that for the ki- the decay rate of Ek . netic energy Ek(t), even in simulations where the initial ͑ ͒ Figures 9–12 and Table III show the decomposition of fluctuating u and b fields are similar Fig. 8 . In all cases ϵ ϩ ϩ ϭ the total energy, Et Ek Eb Ei into its normalized compo- investigated, when Re␴ 1, this quantity is rapidly driven N N N from its initial values ͑0.125 and 0.05556͒ to essentially nents Ek , Eb , and Ei . Note that Figs. 9–12 are grouped in zero. This result has not been reported in the literature, per- a way that allows easy assessment of the effects of magnetic haps because most fundamental studies have focused on ap- Reynolds number. Figure 13 contains the same information, plications where Re␴ is two orders of magnitude larger but the plots are organized to allow easy comparison of the (Re␴ϭ250). In general, the cases with Re␴ϭ250 show evolution pattern of the energy components for each case. weaker decay rates than do the cases with Re␴ϭ100. For f1 We see in Fig. 9 that with normalization, a monotonic effect and ␪ϭ45°, the magnetic energy decays continuously with of Re␴ on the kinetic energy can be observed, thereby show- time from its initial value of 0.125. When ␪ϭ0°, the f1 cases ing a consistent Re␴ effect on the kinetic energy across the show a rapid jump of the magnetic energy from 0.125 to 0.14 various cases investigated. It can also be observed that dur- N before beginning a continuous, nonlinear decay with time. ing most of the transient, the percent contribution (Ek )ofEk

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FIG. 18. Energy spectra at tϭ0.001, 1.0, and 2.0 for f1-45-1.

͑ ͒ to Et decreases with increasing Re␴ Fig. 9 while that of Eb the energy decreases continuously downscale, in an approxi- shows the opposite trend ͑Fig. 10͒. Hence, and as Fig. 11 and mately monotonic fashion, although with some exceptions in ϭ Table III show, the intermediate Re␴ calculations ͑i.e., Re␴ the Pm profile when Re␴ 1. The observations for higher Re␴ ϭ100) appear to give the largest contribution of the internal cases are similar to those in GM1. At low wave numbers, the tendency is for larger spectral energy with decreasing Re␴ . energy Ei to the total energy. This suggests an interaction between Re␴ and the level of density fluctuation ͑or com- However, the spectral decay rate with Re␴ tends to go in the pressibility͒ in the flow. The data also show that, for the opposite direction, at the large to intermediate scales ͑low to ␬ ͒ same ␪ and Re␴ , the f3 cases lead to a relatively high con- intermediate values : the higher Re␴ cases tend to contain tribution of Ek compared to the f1 cases, whereas f3 leads to more energy. The results show that the f3 cases contain more a weaker contribution of Eb . Note that Et decreases with energy at the large scales compared to the f1 cases. Thus, the increasing Re␴ ͑Fig. 12͒, except for the f1-45 cases, where initial relative intensity of the u and b fluctuations affect the the Et for the three Re␴ calculations are of comparable order partitioning of the energy into the spectral modes. The rela- but do not show a definite trend. The Ei enhancement at tive contribution of the three energy modes can also be ob- ͑ Re␴ϭ100 ͑see comments below͒ seems to be responsible for served in the spectra. The kink in the Pm profile around t ϭ ϭ ͒ this observation. For f1-0, Et decreases with Re␴ by a factor 10 when Re␴ 1 is discussed below. That is, at the low of over 2, whereas for f3-45 and f3-0 the ratio is over 3 and and intermediate wave numbers, Pm(k) has significantly ͑ ͒ 7, respectively. Hence, the relative initial intensity of the smaller amplitudes compared to Pk(k)orPi(k) Fig. 14 , flow and magnetic fields determines the dependence on Re␴ . whereas it has a significant contribution at the larger Re␴ ␬ ͑ ͒ The spectra of the kinetic energy, Pk( ), internal energy, values Figs. 15 and 16 . Concerning Pk(k) and Pi(k), the ␬ ␬ Pi( ), and magnetic energy, Pm( ), are shown in Figs. 14– internal energy becomes increasingly significant relative to 19. Some general observations can made, such as the ab- the kinetic energy, as the high wave number end of the spec- sence of a large scale region where the energy increases with trum is approached, such that the two are approximately the wave number, ␬, as in the familiar ␬ϩ2 profile.42 Thus, equipartitioned. This is consistent with the results in GM1.

FIG. 19. Energy spectra at tϭ0.001, 1.0, and 2.0 for f1-45-250.

Re␴ does not affect the initial conditions. However, as time averaged values, those in Figs. 15 and 17 are for the spectra advances, and at the low wave number end of the spectra, the at the specified time steps. intermediate Re␴ calculations show significant levels of in- Figures 18 and 19 have been provided to explain the ͑ ͒ ternal energy relative to the kinetic energy ͑Fig. 15͒. That is, nonmonotonic kink behavior in the Pm(k) profile when ϭ the difference between Pk(k) and Pi(k) at the low wave Re␴ 1. The figures show the short time behavior (t numbers, where the amplitudes are significant, is smallest ϭ0.001, 1, and 2͒. The spectra at tϭ0.001 are clearly those ͱ when Re␴ϭ100, and decreases even further with time. Note of the initial tophat profile in 1рkр 12. Note that the in- that this effect is more pronounced for the f3-45 case ͑Fig. ternal energy spectra do not have the tophat shape because 15͒ compared to the f1-45 case ͑Fig. 17͒. The slow relaxation the initial density field was not filtered with this distribution. ͑or the persistence͒ of the imposed kinetic energy in the In any case, the kinetic energy spectra relax into fairly mono- f1-45 case seems to be responsible for the difference. While tonic distributions early in the transient (tϭ1). The same is the real reason for this relaxation behavior is not entirely true for the magnetic energy spectra when Re␴ϭ100 or 250 clear, it may be due to lower amplification ͑decay͒ rates that ͑Fig. 19͒. For the Re␴ϭ1 case, it appears that a remnant of ͑ are usually associated with small amplitudes, from the view the tophat profile in Pm(k) is transported downscale i.e., to point of hydrodynamic stability.40 ͑See Table II, cases III and the high wave number end of the spectrum͒ as time VIII for the initial kinetic energy levels.͒ Note that, whereas progresses, rather than being ‘‘smeared,’’ leading to the kink ͑ ͒ Fig. 11 or Table III shows the most enhancement of the in the Pm(k) profile at this Re␴ value. internal energy mode by f1-0-100, followed by f1-45-100, Figures 20͑a͒–20͑d͒ are typical spectra of UT and UL for the spectra plots in Figs. 15 and 17 show that the case f3-45- f1-45-1 ͓͑a͒ and ͑b͔͒ and f3-45-1 ͓͑c͒ and ͑d͔͒ at two instants 100 has higher levels of internal energy. These two sets of (tϭ5,30) during the relaxation process. The effects of Re␴ data cannot really be compared because, whereas the data in on these quantities ͑not shown͒ are not as strong as on the Fig. 11 are one-point, having been obtained as spatially respective energies, again suggesting significant interaction

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T L ϭ ͑ ͒ ͑ ͒ FIG. 20. Spectra of U (Pinc(k)) and U (Pcom(k)) at t 5 and 30 for f1-45-1 a,b and f3-45-1 c,d .

between density ␳ and the magnetic Reynolds number Re␴ . with time, monotonically for Re␴ϭ250 but in an oscillatory Finally, as mentioned in GM1, it is difficult, and probably fashion for Re␴ϭ1 ͓Fig. 21͑b͔͒. The tendency toward nega- inappropriate, to assign a steady inertial spectral index to the tive correlation ͑with time evolution͒ as observed for f1-0- computed spectra because of ongoing nonlinear development 250 is not present for f3-0-250. ␪ϭ and limited bandwidth. The Rub results for 45° are significantly different Figures 21͑a͒–21͑d͒ describe the evolution of the cross from those for ␪ϭ0°, the most pronounced being in the re- correlation, Rub : sults for Re␴ϭ1, for both f1 and f3. For this case ͑i.e., ␪ϭ ͒ ϭ ͗u"b͘ 45° , when Re␴ 1, Rub drops off rapidly from its initial R ϭ value of around 1.0, to zero at approximately tϭ0.5 ͑f1͒ and ub ͕͉͗ ͉2͉͗͘ ͉2͖͘1/2 u b tϭ0.4 ͑f3͒. At the higher Re␴ values (Re␴ϭ100,250), the for cases I–XII in Tables I and II. Rub represents the normal- unity value of Rub is maintained for f1-45. For f3-45, it drops ized cross correlation ͑cross helicity͒ between the u and b to between 0.4 and 0.75. fields. That is, an indicator of whether these fields increase or In summary, the most dramatic results for Rub pertain to decrease together, or a decrease in one field is accompanied the fast decay to zero of the correlation when Re␴ϭ1, inde- by an increase in the other. In the initial field, ␪ determines pendent of the initial correlations and the initial relative in- this correlation, as discussed earlier in this paper. Whether an tensities of the flow and magnetic fields. That is, the two initially high correlation remains high depends on the initial fields tend to be uncorrelated when the magnetic Reynolds relative intensities of the two fields. What we see in Fig. 21 number is low. By virtue of the homogeneous nature of the is that ␪ϭ0° not only gives a low correlation at the initial flow and magnetic fields, we can conclude that the fields are time but that the correlation remains weak during the evolu- also statistically independent at the lowest Re␴ value inves- tion of the flow. When ␪ϭ0° and the initial intensities of the tigated, since ͗u"b͘Ϸ͗u͗͘b͘Ϸ0.0. In this case, the role of the fields are comparable ͑i.e., f1-0-1, f1-0-100, and f1-0-250͒, magnetic field is passive, and it is merely convected by the the small correlation there is at the initial instant goes to zero velocity field. In nonhomogeneous velocity and magnetic

FIG. 21. Evolution of the normalized cross helicity, Rub . Refer to the text and Table I for the various cases deﬁned by the legend.

fields, the two fields are more likely to be statistically depen- another demonstration that our calculations are probably dent, even if they are uncorrelated and Re␴ϭ1. Note that to very accurate. Note that the existence or otherwise of a mean maintain a high correlation of the fields during evolution, we field B0 does not affect this conclusion. need to have a high correlation in the initial field and the → The rapid decay of Rub when Re␴ 0 can also be ex- initial relative intensity of the fluctuations must be compa- plained, as follows. As the figures show, the b field decays rable, as in the f1 cases. The Re␴ values must also be large rapidly to attain essentially constant values after the initial within the range used in this paper. transient. On the other hand, the u field continues to decrease The fast temporal decay of the magnetic energy E for b with time. Hence, the two fields are uncorrelated, so that Re␴→0 can be explained if we consider the linearized sys- ͗u"b͘ϭ0, as in the DNS results. tem It is pointed out that the present DNS results for Re␴ b 1 ϭ1 can be used to assess the accuracy of both theץ i ϭ ϩ ⌬ Bojui, j bi . 22 34 t Re␴ quasistatic and the quasilinear approximations. Theseץ simplifications have been studied for incompressible flows; In Fourier space, this equation has the solution the present results can be used to investigate their validity for compressible flows.13 Note that the quasistatic model may 2 t ͑ ͒ϭ ͑ ͒ Ϫ͑k /Re␴͒tϩ ͵ ␶ ͑ ␶͒ not be appropriate for investigating decaying turbulence be- bi k,t bi k,0 e i d k jBojui k, 0 cause of the crucial dependence of the evolution on the ini- Ϫ͑ 2 ͒͑ Ϫ␶͒ tial conditions. A modally forced calculation might be more ϫe k /Re␴ t , acceptable in a DNS work that uses the quasistatic approxi- ͑ 11 which shows a rapid decay of b and Eb) with time for small mation, as in Zikanov and Thess. The quasilinear model is values of Re␴ , as predicted by the DNS calculations. This is more acceptable is this regard, as long as Re␴ is very small.

Downloaded 03 Dec 2005 to 128.112.87.66. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp 2120 Phys. Fluids, Vol. 16, No. 6, June 2004 F. Ladeinde and D. V. Gaitonde

VII. CONCLUDING REMARKS the majority of cases investigated. A spectral plot also sup- ports the unexpected behavior, and calculations were re- We have carried out direct simulation of MHD turbu- peated to confirm the observations. A precise explanation is lence in this paper for the purpose of investigating the effects not available at the moment, although we are currently of the magnetic Reynolds number. Most previous fundamen- studying the energy exchange dynamics between kinetic, tal works on the subject have used conditions or assump- magnetic, and internal components. The procedure followed tions, such as the quasistatic approximations, incompressibil- is conceptually similar to that in Tennekes and Lumley46 for ity, large values of the magnetic Reynolds number, the the role of pressure strain in energy exchange. The details are Orszag–Tang initial conditions, and spectral forcing, that different, however. Because the exercise is a major one, in- make them inappropriate for our interest. We study the ef- volving both theoretical and parametric studies with different fects of magnetic Reynolds number in compressible simula- values of Re␴ , f u , f b , etc., the results from it are not avail- tions that use the full MHD formulation. The initial fields are able for the present paper. carefully determined, with particular emphasis on the rela- An interesting result for Rub pertains to the fast decay to tive intensities of the fluctuating velocity and magnetic fields zero of this quantity when Re␴ϭ1, independent of the initial and the initial cross helicity. The study that most closely correlations and the initial relative intensities of the flow and relates to the present one is GM1. We used the results in magnetic fields. That is, the two fields tend to be uncorre- GM1 to validate our procedure, which differs significantly lated when the magnetic Reynolds number is low so that from that in GM1. However, the details of the initial condi- they are statistically independent at this Re␴ value. In this tions and the parameter ranges investigated in the present case, the role of the magnetic field is passive, and is merely paper are completely different from those in GM1. There- convected by the velocity field. Note that, to maintain a high fore, the details of the results we present in this paper cannot a correlation between the fields during evolution, the corre- be compared directly to those in GM1. There are some quali- lation must be high in the initial field and the initial relative tative agreements in the few cases where the parameters are intensity of the fluctuations must be comparable. The Re␴ of comparable order of magnitude. values must also be large ͑within the range used in this pa- To our knowledge, some of the results reported in this per͒ for the fields to be correlated. paper are new. The studies suggest that the effects of magnetic field on the evolution of the turbulence energy is neg- ACKNOWLEDGMENTS ligible at the lowest magnetic Reynolds number investigated and that at the higher Re␴ values, when magnetic effects are This work was partially supported by the National Re- important, the magnetic field tends to accelerate the decay of search Council ͑NRC͒ Summer Faculty Program and the Air the turbulence energy relative to non-MHD flows. The nature Vehicles Directorate of the Air Force Research Laboratory at of this enhancement in decay rate depends on the particular Wright–Patterson Air Force Base, Dayton, OH. The contri- Re␴ values, the relative initial intensity of the u and b fields, bution of our colleagues Dr. Stavros Kassinos and Dr. X. Cai and the initial correlation between the fields. We have ob- is gratefully acknowledged. The first author ͑F.L.͒ would like 2 served a similar evolution pattern for the quantities M s , Re, to express his appreciation to Dr. Sanjoy Ghosh for the nu- T L Ek(t), and U and for Ei and U , and have provided an merous discussions during the course of this work. Some of explanation for the observation. In all cases investigated, the calculations would have been more difficult to perform, when Re␴ϭ1, the magnetic energy is rapidly driven from its if not impossible, without the hints provided by Dr. Ghosh. initial values to essentially a value of zero very early in the Vivian Shao helped with the plots. transient. Higher Re␴ values significantly slow down the magnetic energy decay rates. We have also seen that the E b 1 decay mode is less sensitive to the initial correlation between J. B. Taylor, ‘‘Relaxation of toroidal plasma and generation of reverse magnetic field,’’ Phys. Rev. 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