Finite Reynolds Number Effects Research Articles

Dual scaling and the n-thirds law in grid turbulence

A dual scaling of the turbulent longitudinal velocity structure function $\overline {{(\delta u)}^n}$ , i.e. a scaling based on the Kolmogorov scales ( $u_K$ , $\eta$ ) and another based on ( $u'$ , $L$ ) representative of the large scale motion, is examined in the context of both the Kármán–Howarth equation and experimental grid turbulence data over a significant range of the Taylor microscale Reynolds number $Re_\lambda$ . As $Re_\lambda$ increases, the scaling based on ( $u'$ , $L$ ) extends to increasingly smaller values of $r/L$ while the scaling based on ( $u_K$ , $\eta$ ) extends to increasingly larger values of $r/\eta$ . The implication is that both scalings should eventually overlap in the so-called inertial range as $Re_\lambda$ continues to increase, thus leading to a power-law relation $\overline {{(\delta u)}^n} \sim r^{n/3}$ when the inertial range is rigorously established. The latter is likely to occur only when $Re_\lambda \to \infty$ . The use of an empirical model for $\overline {{(\delta u)}^n}$ , which complies with $\overline {{(\delta u)}^n} \sim r^{n/3}$ as $Re_\lambda \to \infty$ , shows that the finite Reynolds number effect may differ between even- and odd-orders of $\overline {{(\delta u)}^n}$ . This suggests that different values of $Re_\lambda$ may be required between even and odd values of $n$ for compliance with $\overline {{(\delta u)}^n} \sim r^{n/3}$ . The model describes adequately the dependence on $Re_\lambda$ of the available experimental data for $\overline {{(\delta u)}^n}$ and supports indirectly the extrapolation of these data to infinitely large $Re_\lambda$ .

Higher-order statistics and intermittency of a two-fluid Hall–Vinen–Bekharevich–Khalatnikov quantum turbulent flow

The Hall–Vinen–Bekharevich–Khalatnikov (HVBK) model is widely used to numerically study quantum turbulence in superfluid helium. Based on the two-fluid model of Tisza and Landau, the HVBK model describes the normal (viscous) and superfluid (inviscid) components of the flow using two Navier–Stokes type of equations, coupled through a mutual friction force term. We derive transport equations for the third-order moments for each component of velocity involving the fourth-order moments, which are classical probes for internal intermittency at any scale, and revealing the probability of rare and strong fluctuations. Budget equations are assessed through direct numerical simulations of the HVBK flow. We simulate a forced homogeneous isotropic turbulent flow with Reynolds number of the normal fluid (based on Taylor's microscale) close to 100. Values from 0.1 to 10 are considered for the ratio between the normal and superfluid densities. For these flows, an inertial range is not discernible and the restricted scaling range approach is used to take into account the finite Reynolds number (FRN) effect. We analyse the importance of each term in budget equations and emphasize their role in energy exchange between normal and superfluid components. Some interesting features are observed: (i) transport and pressure-related terms are dominant, similarly to single-fluid turbulence; and (ii) the mathematical signature of the FRN effect is weak despite the low value of the Reynolds number. The flatness of the velocity derivatives is finally studied through the transport equations and their limit for very small scales, and it is shown to gradually increase for lower and lower temperatures, for both normal fluid and superfluid. This similarity highlights the strong locking of the two fluids. The flatness factors are also found in reasonable agreement with classical turbulence.

Two-time Lagrangian velocity correlation function for particle pairs in two-dimensional inverse energy-cascade turbulence

We numerically investigate a two-time Lagrangian velocity correlation function (TTLVCF) for particle pairs in two-dimensional energy inverse-cascade turbulence. We consider self similarity of the correlation function by means of incomplete similarity. In this framework, we propose a self-similar form of the correlation function, whose scaling exponents cannot be determined by only using the dimensional analysis based on the Kolmogorov's phenomenology. As a result, the scaling laws of the correlation function can depend on the initial separation. This initial-separation dependency is frequently observed in laboratory experiments and direct numerical simulations of the relative dispersion, which is directly related to the correlation function, at moderate Reynolds numbers. We numerically verify the self-similar form by direct numerical simulations of two-dimensional energy inverse-cascade turbulence. The involved scaling exponents and the dependencies on finite Reynolds number effects are determined empirically. Then, we consider implication of the scaling laws of the correlation function on the relative dispersion, i.e. the Richardson-Obukhov $t^3$ law. Our results suggest a possibility not to recover the Richardson-Obukhov $t^3$ law at infinite Reynolds number.

New insights from inviscid modelling of starting flow separation with roll-up

Abstract

Finite Reynolds number effect and the 4/5 law

Kolmogorov [A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 30, 299 (1941)] formulated a theory of small-scale turbulence (K41), valid at extremely large Reynolds numbers, based on two similarity hypotheses and on an exact result derived from the transport equation for the second-order structure function, known as the 4/5 law. Although K41 was praised for its simplicity and elegance, Kolmogorov [A. N. Kolmogorov, J. Fluid Mech. 13, 82 (1962).] proposed a new refined similarity hypothesis (K62) mainly to account for the effect of the large scales on the small scales. It has been widely interpreted in the literature as a correction to K41 arising from the intermittency of the instantaneous energy dissipation rate ε. In this paper we argue that since K62 retains the 4/5 law, it must satisfy the same constraints as K41, viz., extremely large Reynolds number and flow stationarity. The retention of the 4/5 law is not however consistent with the presence of nonstationarity due to the effect of the large scales, as postulated by K62. A relatively extensive survey of published data shows that, indeed, the 4/5 law has not yet been observed in either experiments or simulations due to the Reynolds number not being sufficiently large. The use of the transport equation for the second-order structure function, together with an empirical model for the Kolmogorov-normalized second-order velocity structure function, confirms that the 4/5 law is established only after this structure function becomes independent of the Reynolds number.Received 25 April 2019DOI:https://doi.org/10.1103/PhysRevFluids.4.084602©2019 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasTurbulenceFluid Dynamics

Can small-scale turbulence approach a quasi-universal state?

Analytical considerations, based on the Navier-Stokes equations, which take into account the finite Reynolds number effect, together with all available experimental laboratory data, confirm a tendency towards the universal predictions of K41 as the Reynolds number continues to increase.

Reynolds number effect on the velocity derivative flatness factor

This paper investigates the effect of a finite Reynolds number (FRN) on the flatness factor ($F$) of the velocity derivative in decaying homogeneous isotropic turbulence by applying the eddy damped quasi-normal Markovian (EDQNM) method to calculate all terms in an analytic expression for $F$ (Djenidi et al., Phys. Fluids, vol. 29 (5), 2017b, 051702). These terms and hence $F$ become constant when the Taylor microscale Reynolds number, $Re_{\unicode[STIX]{x1D706}}$ exceeds approximately $10^{4}$. For smaller values of $Re_{\unicode[STIX]{x1D706}}$, $F$, like the skewness $-S$, increases with $Re_{\unicode[STIX]{x1D706}}$; this behaviour is in quantitative agreement with experimental and direct numerical simulation data. These results indicate that one must first ensure that $Re_{\unicode[STIX]{x1D706}}$ is large enough for the FRN effect to be negligibly small before the hypotheses of Kolmogorov (Dokl. Akad. Nauk SSSR, vol. 30, 1941a, pp. 301–305; Dokl. Akad. Nauk SSSR, vol. 32, 1941b, pp. 16–18; J. Fluid Mech., vol. 13, 1962, pp. 82–85) can be assessed unambiguously. An obvious implication is that results from experiments and direct numerical simulations for which $Re_{\unicode[STIX]{x1D706}}$ is well below $10^{4}$ may not be immune from the FRN effect. Another implication is that a power-law increase of $F$ with respect to $Re_{\unicode[STIX]{x1D706}}$, as suggested by the Kolmogorov 1962 theory, is not tenable when $Re_{\unicode[STIX]{x1D706}}$ is large enough.

Finite Reynolds number effect on the scaling range behaviour of turbulent longitudinal velocity structure functions

The effect of large-scale forcing on the second- and third-order longitudinal velocity structure functions, evaluated at the Taylor microscale $r=\unicode[STIX]{x1D706}$, is assessed in various turbulent flows at small to moderate values of the Taylor microscale Reynolds number $R_{\unicode[STIX]{x1D706}}$. It is found that the contribution of the large-scale terms to the scale by scale energy budget differs from flow to flow. For a fixed $R_{\unicode[STIX]{x1D706}}$, this contribution is largest on the centreline of a fully developed channel flow but smallest for stationary forced periodic box turbulence. For decaying-type flows, the contribution lies between the previous two cases. Because of the difference in the large-scale term between flows, the third-order longitudinal velocity structure function at $r=\unicode[STIX]{x1D706}$ differs from flow to flow at small to moderate $R_{\unicode[STIX]{x1D706}}$. The effect on the second-order velocity structure functions appears to be negligible. More importantly, the effect of $R_{\unicode[STIX]{x1D706}}$ on the scaling range exponent of the longitudinal velocity structure function is assessed using measurements of the streamwise velocity fluctuation $u$, with $R_{\unicode[STIX]{x1D706}}$ in the range 500–1100, on the axis of a plane jet. It is found that the magnitude of the exponent increases as $R_{\unicode[STIX]{x1D706}}$ increases and the rate of increase depends on the order $n$. The trend of published structure function data on the axes of an axisymmetric jet and a two-dimensional wake confirms this dependence. For a fixed $R_{\unicode[STIX]{x1D706}}$, the exponent can vary from flow to flow and for a given flow, the larger $R_{\unicode[STIX]{x1D706}}$ is, the closer the exponent is to the value predicted by Kolmogorov (Dokl. Akad. Nauk SSSR, vol. 30, 1941a, pp. 299–303) (hereafter K41). The major conclusion is that the finite Reynolds number effect, which depends on the flow, needs to be properly accounted for before determining whether corrections to K41, arising from the intermittency of the energy dissipation rate, are needed. We further point out that it is imprudent, if not incorrect, to associate the finite Reynolds number effect with a consequence of the modified similarity hypothesis introduced by Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82–85) (K62); we contend that this association has misled the vast majority of post K62 investigations of the consequences of K62.

Turbulence in a box: quantification of large-scale resolution effects in isotropic turbulence free decay

The effects of the finiteness of the physical domain over the free decay of homogeneous isotropic turbulence are explored in the present article. Saturation at the large scales is investigated by the use of theoretical analysis and eddy-damped quasi-normal Markovian calculations. Both analyses indicate a strong sensitivity of the large-scale features of the flow to saturation and finite Reynolds number effects. This aspect plays an important role in the general lack of agreement between grid turbulence experiments and numerical simulations. On the other hand, the statistical quantities associated with the behaviour of the spectrum in the inertial region are only marginally affected by saturation. These results suggest new guidelines for the interpretation of experimental and direct numerical simulation studies.

Small scale turbulence and the finite Reynolds number effect

Failure to recognize the importance of the finite Reynolds number effect on small scale turbulence has, by and large, resulted in misguided assessments of the first two hypotheses of Kolmogorov [“Local structure of turbulence in an incompressible fluid for very large Reynolds numbers,” Dokl. Akad. Nauk SSSR 30, 299–303 (1941)] or K41 as well as his third hypothesis [A. N. Kolmogorov, “A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number,” J. Fluid Mech. 13, 82–85 (1962)] or K62. As formulated by Kolmogorov, all three hypotheses require local isotropy to be valid and the Reynolds number to be very large. In the context of the first hypothesis, there is now strong evidence to suggest that this requirement can be significantly relaxed, at least for dissipative scales and relatively low order moments of the velocity structure function. As the scale increases, the effect of the large scale motion on these moments becomes more prominent and higher Reynolds numbers are needed before K41 and K62 can be tested unambiguously.

Relationship between the energy dissipation function and the skin friction law in a turbulent channel flow

Integrals of the mean and turbulent energy dissipation rates are examined using direct numerical simulation (DNS) databases in a turbulent channel flow. Four values of the Kármán number ($h^{+}=180$, 395, 640 and 1020;$h$is the channel half-width) are used. Particular attention is given to the functional$h^{+}$dependence by comparing existing DNS and experimental data up to$h^{+}=10^{4}$. The logarithmic$h^{+}$dependence of the integrated turbulent energy dissipation rate is established for$300\leqslant h^{+}\leqslant 10^{4}$, and is intimately linked to the logarithmic skin friction law,viz.$U_{b}^{+}=2.54\ln (h^{+})+2.41$($U_{b}$ is the bulk mean velocity). This latter relationship is established on the basis of energy balances for both the mean and turbulent kinetic energy. When$h^{+}$is smaller than 300, viscosity affects the integrals of both the mean and turbulent energy dissipation rates significantly due to the lack of distinct separation between inner and outer regions. The logarithmic$h^{+}$dependence of$U_{b}^{+}$is clarified through the scaling behaviour of the turbulent energy dissipation rate$\overline{{\it\varepsilon}}$in different parts of the flow. The overlap between inner and outer regions is readily established in the region$30/h^{+}\leqslant y/h\leqslant 0.2$for$h^{+}\geqslant 300$. At large$h^{+}$(${\geqslant}$5000) when the finite Reynolds number effect disappears, the magnitude of$\overline{{\it\varepsilon}}y/U_{{\it\tau}}^{3}$approaches 2.54 near the lower bound of the overlap region. This value is identical between the channel, pipe and boundary layer as a result of similarity in the constant stress region. As$h^{+}$becomes large, the overlap region tends to contribute exclusively to the$2.54\ln (h^{+})$dependence of the integrated turbulent energy dissipation rate. The present logarithmic$h^{+}$dependence of$U_{b}^{+}$is essentially linked to the overlap region, even at small$h^{+}$.

Modelling of the subgrid scale wrinkling factor for large eddy simulation of turbulent premixed combustion

We propose a model for assessing the unresolved wrinkling factor in the large eddy simulation of turbulent premixed combustion. It relies essentially on a power-law dependence of the wrinkling factor on the filter size and an original expression for the ‘active’ corrugating strain rate. The latter is written as the turbulent strain multiplied by an efficiency function that accounts for viscous effects and the kinematic constraint of Peters. This yields functional expressions for the fractal dimension and the inner cut-off length scale, the latter being (i) filter-size independent and (ii) consistent with the Damköhler asymptotic behaviours at both large and small Karlovitz numbers. A new expression for the wrinkling factor that incorporates finite Reynolds number effects is further proposed. Finally, the model is successfully assessed on an experimental filtered database.

Geometrical properties of turbulent premixed flames and other corrugated interfaces.

This study focuses on the geometrical properties of turbulent flame fronts and other interfaces. Toward that end, we use an original tool based on proper orthogonal decomposition (POD), which is applied to the interface spatial coordinates. The focus is mainly on the degree of roughness of the flame front, which is quantified through the scale dependence of its coverage arclength. POD is first validated by comparing with the caliper technique. Fractal characteristics are extracted in an unambiguous fashion using a parametric expression which appears to be impressively well suited for representing Richardson plots. Then it is shown that, for the range of Reynolds numbers investigated here, the scale-by-scale contribution to the arclength does not comply with scale similarity, irrespectively of the type of similarity which is invoked. The finite ratios between large and small scales, referred to as finite Reynolds number effects, are likely to explain this observation. In this context, the Reynolds number that ought to be achieved for a proper inertial range to be discernible, and for scale similarity to be likely to apply, is calculated. Fractal characteristics of flame folding are compared to available predictions. It is confirmed that the inner cutoff satisfactorily correlates with the Kolmogorov scale while the outer cutoff appears to be proportional to the integral length scale. However, the scaling for the fractal dimension is much less obvious. It is argued that much higher Reynolds numbers have to be reached for drawing firm statements about the evolution (or constancy) of the fractal dimension with respect to flame and flow parameters. Finally, a heuristic phenomenology of corrugated interfaces is highlighted. The degree of generality of the latter phenomenology is confirmed by comparing the folding of different interfaces including a turbulent-nonturbulent interface, a liquid jet destabilized by a surrounding air jet, a cavitating flow, and an isoscalar evolving in a turbulent medium. The latter outcome is likely to have strong implications for modeling the corrugation of turbulent interfaces occurring in many physical situations.

Pressure statistics in self-similar freely decaying isotropic turbulence

AbstractThe time evolution of pressure statistics in freely decaying homogeneous isotropic turbulence (HIT) is investigated via eddy-damped quasi-normal Markovian (EDQNM) computations. The present results show that the time decay rate of pressure-based statistical quantities, such as pressure variance and pressure gradient variance, are sensitive to the breakdown of permanence of large eddies. New formulae for the associated time-decay exponents are proposed, which extend previous relations proposed in Lesieur, Ossia & Metais (Phys. Fluids, vol. 11, 1999, p. 1535). Particular attention is paid to finite-Reynolds-number (FRN) effects on the pressure spectrum and pressure statistics. The results also suggest that $R{e}_{\lambda } = O(1{0}^{4} )$ must be considered to observe a one-decade inertial range in the pressure spectrum with Kolmogorov $- 7/ 3$ scaling. This threshold value is larger than almost all existing direct numerical simulation (DNS) and experimental data, justifying the discussion about other possible scaling laws. The $- 5/ 3$ slope reported in some DNS data is also recovered by the EDQNM model, but it is observed to be a low-Reynolds-number effect. Another important result is that FRN effects yield a departure from asymptotic theoretical behaviours which appear similar to some effects attributed to intermittency by most authors. This is exemplified by the ratio between pressure-based and velocity-based Taylor microscales. Therefore, high-Reynolds-number DNS or experiments such that $R{e}_{\lambda } = O(1{0}^{4} )$ would be required in order to remove FRN effects and to analyse pure intermittency effects.

Spectral approach to finite Reynolds number effects on Kolmogorov’s 4/5 law in isotropic turbulence

The Kolmogorov’s 4/5 law is often considered as the sole exact relationship of inertial range statistics. Its asymptotic character, however, has been evidenced, investigating the finite Reynolds number (FRN) effect for the third-order structure function S3(r) (e.g., for longitudinal velocity increments with r separation length) using variants of the Kármán-Howarth equation in physical space. Similar semi-empirical fits were proposed for the maximum of the normalized structure function, C3 = −maxrS3(r)/(ɛr), expressing C3 − 4/5 as a power law of the Taylor-based Reynolds number. One of the most complete studies in this domain is by Antonia and Burratini [J. Fluid Mech. 550, 175 (2006)]. Considering that these studies are based on a model for the unsteady second-order structure function S2(r,t), with no explicit model for the third-order structure function itself, we propose to revisit the FRN effect by a spectral approach, in the line of Qian [Phys. Rev. E 55, 337 (1997), Phys. Rev. E 60, 3409 (1999)]. The spectral transfer term T(k,t), from which S3(r,t) is derived by an exact quadrature, is directly calculated by solving the Lin equation for the energy spectrum E(k,t), closed by a standard triadic (or three-point) theory, here Eddy Damped Quasi Normal Markovian. We show that the best spectral approach to the FRN effect is found by separately investigating the negative (largest scales) and positive (smaller scales) bumps of the transfer term, and not only by looking at the maximum of the spectral flux or maxk∫k∞T(p,t)dp→ɛ. In the forced case, previous results are well reproduced, with Reynolds numbers as high as Reλ = 5 000 to nearly recover the 4/5 value. In the free decay case, the general trend is recovered as well, with an even higher value of Reλ = 50 000, but the EDQNM plots are systematically below those in Antonia and Burattini [J. Fluid Mech. 550, 175 (2006)]. This is explained by the sensitivity to initial data for E(k) in solving the Lin equation at moderate Reynolds numbers. Accordingly, an ad hoc initialization yields results consistent with the experimental spectrum measurements of Comte-Bellot and Corrsin [J. Fluid Mech. 48(2), 273 (1971)], from which S3(r) are recalculated. Present results show that the dispersion observed in existing data at low Reynolds number may be due to sensitivity to initial spectrum shape, a feature of the flow which is not under control in most of laboratory experiments.

Reynolds number effect on the velocity increment skewness in isotropic turbulence

Second and third order longitudinal structure functions and wavenumber spectra of isotropic turbulence are computed using the eddy-damped quasi-normal Markovian model (EDQNM) and compared to results of the multifractal formalism. It is shown that both the multifractal model and EDQNM give power-law corrections to the inertial range scaling of the velocity increment skewness. For the multifractal formalism, this is an intermittency correction that persists at any high Reynolds number. For EDQNM, this correction is a finite Reynolds number effect, and it is shown that very high Reynolds numbers are needed for this correction to become insignificant with respect to intermittency corrections. Furthermore, the two approaches yield realistic behavior of second and third order statistics of the velocity fluctuations in the dissipative and near-dissipative ranges. Similarities and differences are highlighted, in particular, the Reynolds number dependence.

An equation of motion for particles of finite Reynolds number and size

In order to simulate the motion of bubbles, drops, and particles, it is often important to consider finite Reynolds number effects on drag, lift, torque, and history force. Herein, an equation of motion is developed for spherical particles with a no-slip surface based on theoretical analysis, experimental data, and surface-resolved simulations. The equation of motion is then extended to account for finite particle size. This extension is critical for particles which will have a size significantly larger than the grid cell size, particularly important for bubbles, and low-density particles. The extension to finite particle size is accomplished through spatial-averaging (both volume-based and surface-based) of the continuous flow properties. This averaging is consistent with the Faxen limit for solid spheres at small Reynolds numbers and added mass and fluid stress forces at inviscid limits. The finite Re p corrections are shown to have good agreements with experiments and resolved-surface simulations. The finite size corrections are generally fourth-order accurate and an order of magnitude more accurate than point-force expressions (which neglect quadratic and higher spatial gradients) for particles with size on the order of the gradient length-scales. However, further work is needed for more quantitative assessment of the truncation terms and the overall model robustness and accuracy in complex flows.

On the anomalous behavior of the Lagrangian structure function similarity constant inside dense canopies

The choice of the Kolmogorov constant (C0) in Lagrangian Stochastic Models (LSMs) for canopy flows remains a subject of debate and uncertainty. This uncertainty stems from the fact that canopy flows are highly dissipative, lack a well-defined inertial subrange (ISR) in their energy cascade, and in the deeper layers of the canopy, the attenuation of turbulence can amplify finite Reynolds number effects on C0. From the analysis here, it was shown that C0 inside dense canopies is reduced relative to its value in the atmospheric surface layer (ASL) primarily due to wake production (a factor of 5), followed by finite Reynolds number effects (a factor of 1.5 at most). The short-circuiting of the energy cascade tends to increase C0 though not enough to compensate for the other two reductions. These results are qualitatively consistent with theoretical predictions of a reduced C0 with an increased anisotropy and localized acceleration when referenced to a homogeneous isotropic stationary turbulence. Simplified scaling arguments were proposed for each of these three effects and tested using flume experiments. The fact that C0 may vary nonlinearly inside canopies complicates inverse estimates of C0 that use fitting Lagrangian dispersion models (LDMs) to mean concentration measurements. The C0 values inferred from such an approach were shown to be sensitive to the source location (especially inside the canopy) and concentration sampling points. On a positive note, the fact that C0 may vary within the canopy does not require any revisions to the well-mixed condition because LDM are not sensitive to gradients in C0. A phenomenological model that accounts for the vertical variation in C0 as a function of the most elementary flow variables, the mean velocity and canopy adjustment length scale, is proposed but its general applicability remains to be tested.

Nonlinear wave propagation in a gas with algebraic density distribution

The nonlinear plane wave, which propagates in a perfect gas with an algebraic density distribution, is studied. A generalized Burgers equation is derived for a far field of weakly nonlinear waves, using the multiple-scale method. For lossless media, it follows that a sinusoidal wave may evolve into a sawtooth-like wave, or it may not fully into a sawtooth-like wave and in some cases no shock may emerge. For the algebraic density distribution, the evolution of the shock strength in velocity can be classified into five types. The classification diagram can be completely determined by two parameters. Numerical results show that finite-Reynolds number effects may well suppress an enormous increase in velocity due to the inhomogeneity, and that a strongly nonlinear wave can be produced together with intense streaming.

Finite Reynolds number effect on the rheology of a dilute suspension of neutrally buoyant circular particles in a Newtonian fluid

Knowledge of suspension rheology can help in the prediction of its behavior under various flow conditions. Considerable theoretical effort has been made to model the macroscopic properties of suspensions of rigid particles in a Newtonian fluid, but for the most part it has been limited to zero particle Reynolds number. In this paper we investigate the effect of finite particle Reynolds number on the macroscopic properties of suspensions of rigid circular particles in a Newtonian fluid in the dilute limit (no interaction between particles). Two-dimensional numerical simulations are performed. Navier–Stokes equations are solved for the fluid without neglecting any terms. It is seen that the viscosity of the suspension shear thickens. The first normal stress difference is negative and increases in magnitude as the Reynolds number is increased.