Compressible Turbulence Research Articles

Predictive models for near-wall velocity and temperature fluctuations in supersonic wall-bounded turbulence

Predictive models for near-wall velocity and temperature fluctuations in compressible wall-bounded turbulence are developed in the present study based on the model proposed by Marusic et al. (Science, vol. 329 (5988), 2010, pp. 193–196), which incorporates the superposition and amplitude modulation effects of the large-scale motions in the outer region on near-wall turbulence. The density variation is involved in the predictive model for velocity fluctuations to achieve Mach number independence. The predictive model for temperature fluctuations is derived to keep its consistency with the strong Reynolds analogy, in which the modulation effect is supposed to be cast as the quadratic function of the large-scale velocity fluctuations. An algebraic method is proposed to directly determine the modulation coefficients and extract the universal signals. A direct numerical simulation (DNS) of turbulent channel flow at the friction Reynolds number of $1170$ and bulk Mach number of $2.88$ is carried out for parameter calibration and validations. The variances and joint probability density functions of the predicted velocity and temperature fluctuations agree well with the DNS results.

Universal turbulence scaling law −8/3 at fusion implosion

A new interpretation of the known results of simulation of the turbulent regime at the time before the stagnation stage of fusion implosion is stated. For this aim, the universal turbulence energy spectrum obtained by the authors with a scaling law −8/3, which corresponds to the exact solution of one-dimensional Euler equations for the dynamics of a compressible medium, is used. It is stated that the scaling law −8/3 has more relevance in comparison with the Kolmogorov spectrum of −5/3 in the inertial sub-range of scales for the compressible turbulence at this stage of fusion implosion. A possible mechanism for the occurrence of the anisotropic spectrum −8/3 in turbulence associated with hydrodynamic instability of rotation of the medium behind the shock wave front is considered.

The Magnetic Field versus Density Relation in Star-forming Molecular Clouds

Abstract We study the magnetic field to density (B–ρ) relation in turbulent molecular clouds with dynamically important magnetic fields using nonideal three-dimensional magnetohydrodynamic simulations. Our simulations show that there is a distinguishable break density ρ T between the relatively flat low-density regime and a power-law regime at higher densities. We present an analytic theory for ρ T based on the interplay of the magnetic field, turbulence, and gravity. The break density ρ T scales with the strength of the initial Alfvén Mach number  A 0 for sub-Alfvénic (  A 0 < 1 ) and trans-Alfvénic (  A 0 ∼ 1 ) clouds. We fit the variation of ρ T for model clouds as a function of  A 0 , set by different values of initial sonic Mach number  0 and the initial ratio of gas pressure to magnetic pressure β 0. This implies that ρ T, which denotes the transition in mass-to-flux ratio from the subcritical to the supercritical regime, is set by the initial turbulent compression of the molecular cloud.

Refractive index fluctuation spectrum of lightwave propagation in supersonic compressible turbulent flow

When an aircraft flies at supersonic speed, a boundary layer is formed around the vehicle. The boundary layer gradually transitions to compressible turbulence near the aircraft wall. An aero-optical effect occurs when a beam passes through compressible turbulence, potentially causing wavefront distortion. Thus, the refractive index spectrum is very important in aero-optical research. In this paper, the general form of the refractive index function is derived from the velocity structure function of compressible flow. Although the general form of the refractive index spectrum is derived, it cannot be used in practice and can only predict a power law. This power law shows that the slope of the spectrum is related not only to −11/3 but also to −(m+2). In practice, the refractive index spectrum experienced by a propagating beam due to compressible turbulence near an aircraft wall is described by the autocorrelation function of the refractive index, which proves that m = 7/3 in this type of turbulence. In contrast to the traditional turbulent atmospheric spectrum, this study considers not only the temperature spectrum but also the pressure spectrum.

Acceleration statistics of tracer and light particles in compressible homogeneous isotropic turbulence

The accelerations of tracer and light particles (bubbles) in compressible homogeneous isotropic turbulence are investigated by using data from direct numerical simulations up to turbulent Mach number $M_t =1$ . For tracer particles, the flatness factor of acceleration components, $F_a$ , increases gradually for $M_t \in [0.3, 1]$ . On the contrary, $F_a$ for bubbles develops a maximum around $M_t \sim 0.6$ . The probability density function of longitudinal acceleration of tracers is increasingly skewed towards the negative value as $M_t$ increases. By contrast, for light particles, the skewness factor of longitudinal acceleration, $S_a$ , first becomes more negative with the increase of $M_t$ , and then goes back to $0$ when $M_t$ is larger than $0.6$ . Similarly, differences among tracers and bubbles appear also in the zero-crossing time of acceleration correlation. It is argued that all these phenomena are intimately linked to the flow structures in the compression regions close to shocklets.

Simulation of three-dimensional forced compressible isotropic turbulence by a redesigned discrete unified gas kinetic scheme

In this paper, we implemented the Boltzmann-equation-based mesoscopic model, developed recently by Chen et al. [“Inverse design of mesoscopic models for compressible flow using the Chapman–Enskog analysis,” Adv. Aerodyn. 3, 5 (2021)], to simulate three-dimensional (3D) forced compressible isotropic turbulence. In this model, both the Prandtl number and the ratio of bulk to shear viscosity can be arbitrary prescribed. The statistically stationary turbulent flow is driven by a large-scale momentum forcing in the Fourier space, with the internal heating due to the viscous dissipation at small scales being removed by a thermal cooling function. Under the framework of discrete unified gas kinetic scheme (DUGKS), a 3D direct numerical simulation code has been developed, incorporating a generalized Strang-splitting scheme. The weighted essentially non-oscillatory (WENO) scheme is used to increase local spatial accuracy in the reconstruction of particle distribution functions at the cell interface. A 3D discrete particle velocity model with a ninth-order Gauss–Hermite quadrature accuracy is used to ensure accurate evaluation of viscous stress and heat flux in the continuum regime. We simulate forced compressible isotropic turbulence at both low and high turbulent Mach numbers. A direct comparison is performed with the results obtained from a hybrid compact finite difference-WENO scheme solving directly the Navier–Stokes–Fourier system. The comparison validates our DUGKS code and indicates that DUGKS is a reliable and promising tool for simulating forced compressible isotropic turbulence. The work represents a first study to directly simulate forced compressible turbulence by a mesoscopic method based on the Boltzmann equation.

Dynamic tensorial eddy viscosity model: Effects of compressibility and of complex geometry

A previous paper by Cimarelli et al. [“General formalism for a reduced description and modelling of momentum and energy transfer in turbulence,” J. Fluid Mech. 866, 865–896 (2019)] has shown that every decomposition of turbulent stresses is naturally approximated by a general form of tensorial eddy viscosity based on velocity increments. The generality of the formalism is such that it can also be used to give a reduced description of subgrid scalar fluxes. In the same work, this peculiar property of turbulent stresses and fluxes has been dynamically exploited to produce tensorial eddy viscosity models based on the second-order inertial properties of the grid element. The basic idea is that the anisotropic structure of the computational element directly impacts, although implicitly, the large resolved and small unresolved scale decomposition. In the present work, this new class of turbulence models is extended to compressible turbulence. A posteriori analysis of flow solutions in a compressible turbulent channel shows very promising results. The quality of the modeling approach is further assessed by addressing complex flow geometries, where the use of unstructured grids is demanded as in real world problems. Also in this case, a posteriori analysis of flow solutions in a periodic hill turbulent flow shows very good behavior. Overall, the generality of the formalism is found to allow for an accurate description of subgrid quantities in compressible conditions and in complex flows, independent of the discretization technique. Hence, we believe that the present class of turbulence closures is very promising for the applications typical of industry and geophysics.

On the Existence of Fast Modes in Compressible Magnetohydrodynamic Turbulence

Abstract We study the existence and properties of fast magnetosonic modes in 3D compressible MHD turbulence by carrying out a number of simulations with compressible and incompressible driving conditions. We use two approaches to determine the presence of fast modes: mode decomposition based on spatial variations only and spatio-temporal 4D fast Fourier transform (4D FFT) analysis of all fluctuations. The latter method enables us to quantify fluctuations that satisfy the dispersion relation of fast modes with finite frequency. Overall, we find that the fraction of fast modes identified via the spatio-temporal 4D FFT approach in total fluctuation power is either tiny with nearly incompressible driving or ∼2% with highly compressible driving. We discuss the implications of our results for understanding the compressible fluctuations in space and astrophysical plasmas.

A Compressible Turbulence Model for Pressure—Strain

This work focuses on the performance and validation of compressible turbulence models for the pressure-strain correlation. Considering the Launder Reece and Rodi (LRR) incompressible model for the pressure-strain correlation, Adumitroaie et al., Huang et al., and Marzougui et al., used different modeling approaches to develop turbulence models, taking into account compressibility effects for this term. Two numerical coefficients are dependent on the turbulent Mach number, and all of the remaining coefficients conserve the same values as in the original LRR model. The models do not correctly predict the compressible turbulence at a high-speed shear flow. So, the revision of these models is the major aim of this study. In the present work, the compressible model for the pressure-strain correlation developed by Khlifi−Lili, involving the turbulent Mach number, the gradient, and the convective Mach numbers, is used to modify the linear mean shear strain and the slow terms of the previous models. The models are tested in two compressible turbulent flows: homogeneous shear flow and the newly developed plane mixing layers. The predicted results of the proposed modifications of the Adumitroaie et al., Huang et al., and Marzougui et al., models and of its universal versions are compared with direct numerical simulation (DNS) and experiment data. The results show that the important parameters of compressibility in homogeneous shear flow and in the mixing layers are well predicted by the proposal models.

Finite-size evaporating droplets in weakly compressible homogeneous shear turbulence

We perform interface-resolved simulations of finite-size evaporating droplets in weakly compressible homogeneous shear turbulence. The study is conducted by varying three dimensionless physical parameters: the initial gas temperature over the critical temperature $T_{g,0}/T_c$ , the initial droplet diameter over the Kolmogorov scale $d_0/\eta$ and the surface tension, i.e. the shear-based Weber number, $We_{\mathcal {S}}$ . For the smallest $We_{\mathcal {S}}$ , we first discuss the impact on the evaporation rate of the three thermodynamic models employed to evaluate the gas thermophysical properties: a constant property model and two variable-properties approaches where either the gas density or all the gas properties are allowed to vary. Taking this last approach as reference, the model assuming constant gas properties and evaluated with the ‘1/3’ rule is shown to predict the evaporation rate better than the model where the only variable property is the gas density. Moreover, we observe that the well-known Frössling/Ranz-Marshall correlation underpredicts the Sherwood number at low temperatures, $T_{g,0}/T_c=0.75$ . Next, we show that the ratio between the actual evaporation rate in turbulence and the one computed in stagnant conditions is always much higher than one for weakly deformable droplets: it decreases with $T_{g,0}/T_c$ without approaching unity at the highest $T_{g,0}/T_c$ considered. This suggests an evaporation enhancement due to turbulence also in conditions typical of combustion applications. Finally, we examine the overall evaporation rate and the local interfacial mass flux at higher $We_{\mathcal {S}}$ , showing a positive correlation between evaporation rate and interfacial curvature, especially at the lowest $T_{g,0}/T_c$ .

Multiple-Gpu Accelerated High-Order Gas-Kinetic Scheme for Direct Numerical Simulation of Compressible Turbulence

High-order gas-kinetic scheme (HGKS) has become a workable tool for the direct numerical simulation (DNS) of turbulence. In this paper, to accelerate the computation, HGKS is implemented with the graphical processing unit (GPU) using the compute unified device architecture (CUDA). To conduct the much large-scale DNS of turbulence, HGKS also be further upgraded with multiple GPUs using message passing interface (MPI) and CUDA architecture. The benchmark cases for compressible turbulence, including Taylor-Green vortex and turbulent channel flows, are presented to assess the numerical performance of HGKS with Nvidia TITAN RTX and Tesla V100 GPUs. For single-GPU computation, compared with the parallel central processing unit (CPU) code running on the Intel Core i7-9700 with open multi-processing (OpenMP) directives, 7x speedup is achieved by TITAN RTX and 16x speedup is achieved by Tesla V100. For multiple-GPU computation, the computational time of parallel CPU code running on 1024 Intel Xeon E5-2692 cores with MPI is approximately 3 times longer than that of GPU code using 8 Tesla V100 GPUs with MPI and CUDA. Numerical results confirm the excellent performance of multiple-GPU accelerated HGKS for large-scale DNS of turbulence. HGKS in GPU is also compiled with FP32 precision to evaluate the effect of number formats precision. Reasonably, compared to the computation with FP64 precision, the efficiency is improved and the memory cost is reduced with FP32 precision. For turbulent channel flows, difference in long-time statistical turbulent quantities is acceptable between FP32 and FP64 precision solutions. While the obvious discrepancy in instantaneous turbulent quantities can be observed, which shows that FP32 precision is not safe for DNS in compressible turbulence. The choice of precision should depended on the requirement of accuracy and the available computational resources.

Non-Equilibrium Time-Relaxation Kinetic Model for Compressible Turbulence Modeling

For the first time, the non-equilibrium time-relaxation kinetic model (NTRKM) is proposed for compressible turbulence modeling on unresolved grids. Within the non-equilibrium time-relaxation framework, NTRKM is extended in the form of modified Bhatnagar-Gross-Krook model. Based on the first-order Chapman-Enskog expansion, NTRKM connects with the six-variable macroscopic governing equations. The first five governing equations correspond to the conservative laws in mass, momentum and total energy, while the sixth equation governs the evolution of unresolved turbulence kinetic energy Kutke. The unknowns in NTRKM, including turbulent relaxation time and source term, are determined by essential gradient-type assumption and standard dynamic modeling approach. Current generalized kinetic model on unresolved grids consequently offers a profound mesoscopic understanding for one-equation subgrid-scale turbulence kinetic energy Ksgs model in compressible large eddy simulation. To solve NTRKM accurately and robustly, a non-equilibrium gas-kinetic scheme is developed, which succeeds the well-established gas-kinetic scheme for simulating Navier-Stokes equations. Three-dimensional decaying compressible isotropic turbulence and temporal compressible plane mixing layer on unresolved grids are simulated to evaluate the generalized kinetic model and non-equilibrium gas-kinetic scheme. The performance of key turbulent quantities up to second-order statistics confirms that NTRKM is comparable with the widely-used eddy-viscosity Smagorinsky model (SM) and dynamic Smagorinsky model (DSM). Specifically, compared with the DNS solution in temporal compressible plane mixing layer, the performance of NTRKM is much closer with DSM and better than SM. This study provides a workable approach for compressible turbulence modeling on unresolved grids.

ЧИСЛЕННОЕ МОДЕЛИРОВАНИЕ ПРОЦЕССОВ ВЗАИМОВЛИЯНИЯ ВОЛНЫ ГАЗА И ДЕФОРМИРОВАННОЙ ПРЕГРАДЫ В МОДЕЛЬНОМ КАНАЛЕ АВИАЦИОННОГО ДВИГАТЕЛЯ

The aim of the study is to adapt the ANSYS CFX and ANSYS Mechanical engineering analysis systems for numerical calculations of acoustic processes in the model channels of an aircraft engine. Numerical experiments were carried out in the 2FSI (2-way Fluid-Structure Interaction Explained) formulation, which allows taking into account the interaction between the gas and the structure. A mathematical model for gas-dynamic processes is formulated, which is based on the laws of conservation of mass, momentum, energy and is closed by the equations of state of an ideal compressible gas and turbulence. A mathemati-cal model is formulated for estimating the stress-strain state of a structure, taking into account the movement of the partition in the framework of the linear theory of elasticity. Boundary conditions are determined. The task of the study was to evaluate the influence of the choice of the type of partition material on the propagation of a sound wave along the flow to the partition and beyond. As a partition, materials were chosen that differ in the value of the modulus of elasticity and Poisson's ratio, steel, tita-nium and polyethylene. Graphical dependences of the pressure amplitude at six control points are obtained. Graphs of moving partitions depending on the type of material are obtained. Computational experiments were carried out at Perm National Re-search Polytechnic University using the computing resources of the cluster of the Center for High-Performance Computing Sys-tems.

Global hypocoercivity of kinetic Fokker-Planck-Alignment equations

<p style='text-indent:20px;'>In this note we establish hypocoercivity and exponential relaxation to the Maxwellian for a class of kinetic Fokker-Planck-Alignment equations arising in the studies of collective behavior. Unlike previously known results in this direction that focus on convergence near Maxwellian, our result is global for hydrodynamically dense flocks, which has several consequences. In particular, if communication is long-range, the convergence is unconditional. If communication is local then all nearly aligned flocks quantified by smallness of the Fisher information relax to the Maxwellian. In the latter case the class of initial data is stable under the vanishing noise limit, i.e. it reduces to a non-trivial and natural class of traveling wave solutions to the noiseless Vlasov-Alignment equation.</p><p style='text-indent:20px;'>The main novelty in our approach is the adaptation of a mollified Favre filtration of the macroscopic momentum into the communication protocol. Such filtration has been used previously in large eddy simulations of compressible turbulence and its new variant appeared in the proof of the Onsager conjecture for inhomogeneous Navier-Stokes system. A rigorous treatment of well-posedness for smooth solutions is provided. Lastly, we prove that in the limit of strong noise and local alignment solutions to the Fokker-Planck-Alignment equation Maxwellialize to solutions of the macroscopic hydrodynamic system with the isothermal pressure.</p>

Exact mathematical formulas for wall-heat flux in compressible turbulent channel flows

In this paper, several exact expressions for the mean heat flux at the wall ($q_w$) for the compressible turbulent channel flows are derived by using the internal energy equation or the total energy equation. Two different routes, including the FIK method and the RD method, can be applied. The direct numerical simulations data of compressible channel flows at different Reynolds and Mach numbers verify the correctness of the derived formulas. Discussions related to the different energy equations, and different routes are carried out, and we may arrive at the conclusion that most of the formulas derived in the present work are just mathematically ones which generally are lack of clear physical interpretation. They can be used to estimate $q_w$, but might not be suitable for explore the underlying physics.

Asymptotic behaviour at the wall in compressible turbulent channels

The asymptotic behaviour of Reynolds stresses close to walls is well established in incompressible flows owing to the constraint imposed by the solenoidal nature of the velocity field. For compressible flows, thus, one may expect a different asymptotic behaviour, which has indeed been noted in the literature. However, the transition from incompressible to compressible scaling, as well as the limiting behaviour for the latter, is largely unknown. Thus, we investigate the effects of compressibility on the near-wall, asymptotic behaviour of turbulent fluxes using a large direct numerical simulation (DNS) database of turbulent channel flow at higher than usual wall-normal resolutions. We vary the Mach number at a constant friction Reynolds number to directly assess compressibility effects. We observe that the near-wall asymptotic behaviour for compressible turbulent flow is different from the corresponding incompressible flow even if the mean density variations are taken into account and semi-local scalings are used. For Mach numbers near the incompressible regimes, the near-wall asymptotic behaviour follows the well-known theoretical behaviour. When the Mach number is increased, turbulent fluxes containing wall-normal components show a decrease in the slope owing to increased dilatation effects. We observe that $R_{vv}$ approaches its high-Mach-number asymptote at a lower Mach number than that required for the other fluxes. We also introduce a transition distance from the wall at which turbulent fluxes exhibit a change in scaling exponents. Implications for wall models are briefly presented.

Kinetic-energy-flux-constrained model using an artificial neural network for large-eddy simulation of compressible wall-bounded turbulence

Kinetic energy flux (KEF) is an important physical quantity that characterizes cascades of kinetic energy in turbulent flows. In large-eddy simulation (LES), it is crucial for the subgrid-scale (SGS) model to accurately predict the KEF in turbulence. In this paper, we propose a new eddy-viscosity SGS model constrained by the properly modelled KEF for LES of compressible wall-bounded turbulence. The new methodology has the advantages of both accurate prediction of the KEF and strong numerical stability in LES. We can obtain an approximate KEF by the tensor-diffusivity model, which has a high correlation with the real value. Then, using the artificial neural network method, the local ratios between the real KEF and the approximate KEF are accurately modelled. Consequently, the SGS model can be improved by the product of that ratio and the approximate KEF. In LES of compressible turbulent channel flow, the new model can accurately predict mean velocity profile, turbulence intensities, Reynolds stress, temperature–velocity correlation, etc. Additionally, for the case of a compressible flat-plate boundary layer, the new model can accurately predict some key quantities, including the onset of transitions and transition peaks, the skin-friction coefficient, the mean velocity in the turbulence region, etc., and it can also predict the energy backscatters in turbulence. Furthermore, the proposed model also shows more advantages for coarser grids.

Why take the square root? An assessment of interstellar magnetic field strength estimation methods

Context. The magnetic field strength in interstellar clouds can be estimated indirectly from measurements of dust polarization by assuming that turbulent kinetic energy is comparable to the fluctuating magnetic energy, and using the spread of polarization angles to estimate the latter. The method developed by Davis (1951, Phys. Rev., 81, 890) and by Chandrasekhar and Fermi (1953, ApJ, 118, 1137) (DCF) assumes that incompressible magnetohydrodynamic (MHD) fluctuations induce the observed dispersion of polarization angles, deriving B ∝ 1∕δθ (or, equivalently, δθ ∝ MA, in terms of the Alfvénic Mach number). However, observations show that the interstellar medium is highly compressible. Recently, two of us (ST) relaxed the incompressibility assumption and derived instead B ∝ 1/√δθ (equivalently, δθ ∝ MA2). Aims. We explored what the correct scaling is in compressible and magnetized turbulence through theoretical arguments, and tested the assumptions and the accuracy of the two methods with numerical simulations. Methods. We used 26 magnetized, isothermal, ideal-MHD numerical simulations without self-gravity and with different types of forcing. The range of MA and sonic Mach numbers Ms explored are 0.1 ≤ MA ≤ 2.0 and 0.5 ≤ Ms ≤ 20. We created synthetic polarization maps and tested the assumptions and accuracy of the two methods. Results. The synthetic data have a remarkable consistency with the δθ ∝ MA2 scaling, which is inferred by ST, while the DCF scaling failed to follow the data. Similarly, the assumption of ST that the turbulent kinetic energy is comparable to the root-mean-square (rms) of the coupling term of the magnetic energy between the mean and fluctuating magnetic field is valid within a factor of two for all MA (with the exception of solenoidally driven simulations at high MA, where the assumption fails by a factor of 10). In contrast, the assumption of DCF that the turbulent kinetic energy is comparable to the rms of the second-order fluctuating magnetic field term fails by factors of several to hundreds for sub-Alfvénic simulations. The ST method shows an accuracy better than 50% over the entire range of MA explored; DCF performs adequately only in the range of MA for which it has been optimized through the use of a “fudge factor”. For low MA, it is inaccurate by factors of tens, since it omits the magnetic energy coupling term, which is of first order and corresponds to compressible modes. We found no dependence of the accuracy of the two methods on Ms. Conclusions. The assumptions of the ST method reflect better the physical reality in clouds with compressible and magnetized turbulence, and for this reason the method provides a much better estimate of the magnetic field strength over the DCF method. Even in super-Alfvénic cases where DCF might outperform ST, the ST method still provides an adequate estimate of the magnetic field strength, while the reverse is not true.

Effect of heat source on statistics and scaling in compressible homogeneous shear turbulence

The effects of heat sources on the velocity and pressure spectra, Mach number scaling of one-point statistics, and small-scale structures of compressible homogeneous shear turbulence are numerically studied. The dilatational components of flow fields are significantly enhanced by a strong heat source at low turbulent Mach numbers Mt and are dominated by an acoustic mode, leading to a strong acoustic equilibrium between the dilatational velocity and pressure. As the magnitude of the heat source increases, the scaling behaviors of the dilatational components of kinetic energy and dissipation rate change from Mt4 and approach a state that is nearly independent of the turbulent Mach number. Furthermore, a strong heat source has a significant effect on small-scale structures at low turbulent Mach numbers. The conditional probability density functions of the normalized eigenvalues of a strain rate tensor become more dependent on the dilatation owing to the effect of the heat source. For low turbulent Mach numbers with strong heat sources, the ratio of the normalized eigenvalues of the strain rate tensor tends to −1:0:0 and −0.2:0.25:1 in the strong compression and strong expansion regions, respectively, and the dilatational vortex stretching term can significantly enhance the enstrophy production.

Field line random walk in magnetic turbulence

The stochastic behavior of magnetic field lines in turbulence is explored analytically and numerically. This problem is a fundamental aspect of turbulence research but also highly relevant in the theory of energetic particles. In the current paper, previous approaches are reviewed and some simple heuristic arguments are provided helping the reader to understand the reason for the form of analytical results. The importance of the so-called Kubo number in field line random walk theory is also discussed. Furthermore, analytical results for a position-dependent field line diffusion coefficient are provided. For more realistic turbulence configurations, the field line diffusion coefficients are computed numerically. This includes quasi-slab, quasi-2D, two-component, and three-dimensional turbulence. Specific aspects of the field line random walk in each model are also discussed. Results based on a diffusion approximation are compared with numerical results obtained without employing this approximation with the aim to explore its validity and accuracy. Numerical results based on simulations for incompressible and compressible turbulence are also discussed.