On the Turbulence Structure in Supersonic Nozzle Flow | SpringerLink

On the Turbulence Structure in Supersonic Nozzle Flow Somnath Ghosh and Rainer Friedrich

Abstract Effects of extra strain and dilatation rates on the turbulence structure in nozzles with fully developed supersonic pipe flow as inflow condition are investigated by means of DNS and LES using high-order numerical schemes. It is found that weak favorable pressure gradients already strongly inhibit the Reynolds stresses via corresponding changes of production and pressure-strain terms. The results constitute a database for the improvement of turbulence models for compressible flows.

1 Introduction Compressibility effects in simple turbulent shear flows along isothermal walls, like fully developed channel or pipe flow, manifest themselves in terms of mean density and temperature gradients in the near-wall layer and thereby increase the anisotropy of the Reynolds stress tensor. While the peak value of the streamwise Reynolds stress grows with increasing Mach number, the peak values of the other stresses decrease as a consequence of reduced pressure-strain correlations. Since wavepropagation effects are unimportant up to supersonic Mach numbers, solutions of the Poisson equation for the pressure fluctuations by means of a Green function have proven for fully-developed supersonic channel flow [6] that the decrease in mean density from the wall to the channel core is responsible for the decrease of all pressure-strain correlations compared to incompressible flow. Analogous effects hold for pipe flow [7]. Now, the production of the streamwise Reynolds stress declines with increasing Mach number, but scales with the wall shear stress and the local viscosity along the semi-local wall-normal coordinate. The corresponding pressure-strain correlation does not follow this scaling law, and decreases faster with increasing Mach number which explains the increase in streamwise Reynolds stress. There is a lack of knowledge concerning the response of compressible wallbounded turbulence to acceleration in axisymmetric nozzles. The response leads to phenomena which cannot be explained in terms of mean property variations alone. Bradshaw [3, 4] has used the appropriate term ‘complex flows’ to denote flows in which significant pressure gradients and strain rates exist. He has discussed the S. Ghosh · R. Friedrich Fachgebiet Strömungsmechanik, Technische Universität München, Boltzmannstr. 15, 85748 Garching, Germany e-mail: [email protected]; [email protected] 241

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increase in Reynolds stresses by bulk compression and their decrease by bulk expansion in supersonic turbulent boundary layers and has highlighted the need to improve engineering calculation methods. His attempt to account for mean dilatation effects in his empirical Reynolds shear stress equation provides some, but not sufficient improvement in predicting complex compressible flows. In their review article on the physics of supersonic turbulent boundary layers, Spina et al. [14] note a lack of knowledge with respect to the influence of extra rates of strain and trace it back to the scarcity of systematic high-quality measurements. It is the aim of this paper to report new findings about effects of extra strain and dilatation rates on the turbulence structure in supersonic nozzles using wellestablished and accurate numerical methods. The flow configuration chosen is fullydeveloped supersonic pipe flow subjected to gradual acceleration in a nozzle with a cooled isothermal wall. While this is one of the simplest ways to subject a pipe flow to a pressure gradient in an experimental setup, the computation of this kind of flow requires proper specification of inflow and outflow conditions. Large-eddy simulations of these flows using an explicit filtering version of the approximate deconvolution method (ADM) of Stolz and Adams [15] have been undertaken here. A DNS has also been performed to validate the LES data.

2 Computational Details of DNS and LES The governing Navier-Stokes equations are solved in a non-conservative pressurevelocity-entropy form [13] on curvilinear coordinates using 6th order compact central schemes [9] for spatial discretizations in the LES. In the DNS, 5th order compact upwind schemes [1] have been used for the convection terms and 6th order compact central schemes for the molecular transport terms. The flow field is advanced in time in both cases using a 3rd order low-storage Runge-Kutta scheme [16]. ADM, implemented as a single-step explicit filtering approach [10], is applied to treat the interaction between resolved and unresolved scales. Ghosh et al. [7] had performed DNS and LES of supersonic pipe flow with an isothermal wall using the above mentioned discretization and explicit filtering technique and found very good agreement between DNS and LES data concerning correlations that are dominated by large scales. In the present work, fully-developed supersonic turbulent air flow in a pipe serves as inflow condition for the nozzle flow. The walls are kept at constant temperature in both cases. The inlet Mach and friction Reynolds numbers for the nozzle are 1.5 and 245. While the Mach number M is based on the speed of sound at wall temperature and the bulk velocity, the friction Reynolds number Reτ √ is defined using the friction velocity uτ = τw /ρw , the pipe radius R and the kinematic viscosity at the wall, νw (Tw ). The domain length of each configuration (pipe or nozzle) is L = 10R. The axial variation of the flow cross section is calculated using streamtube equations for a given axial pressure distribution. This is done to ensure that the turbulence is subjected to an extended region of nearly constant weak pressure gradient. The magnitude of the acceleration can be judged from the value ∗ of the Clauser parameter β = τδw ddxp¯ (δ ∗ denotes displacement thickness) which has

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an average value of −1.6. The ratio of nozzle radius to pipe radius at the end of the computational domains is 1.58. The number of grid points used to discretize these domains is 64 × 64 × 50 in the LES and 256 × 128 × 91 in the DNS in axial (x), azimuthal (φ) and radial (r) directions. Similar spatial resolutions were shown to be adequate for DNS/LES of turbulent pipe flow at M = 1.5 and Reτ = 245 [7]. The pipe and nozzle simulations are coupled using MPI routines following a procedure similar to that of Schlueter et al. [12]. The concept of characteristics is applied to set inviscid inflow conditions for the spatially developing flows. The incoming characteristics are computed from the periodic pipe flow simulations and are received at every time-step in the nozzle computation. For the viscous terms at the inflow, the axial derivatives are computed on a mixed stencil involving points from the pipe and the nozzle simulations. Partially non-reflecting outflow conditions [11] are used in the subsonic region of the outflow plane. The axial derivatives of the shear stresses parallel to the outflow plane and of the heat flux through the outflow plane are set to zero as conditions on viscous terms. No sponge layer has been used either in the LES or in the DNS. The coupled DNS flow solver has a performance of about 400 Mflops/core on the Altix 4700. The DNS was run using 64 cores and required 138240 CPU-hours till the statistics reached a converged state. A typical LES of this flow is carried out using 8 cores and requires 5760 CPU-hours.

3 Results As was already mentioned in the introduction, mean property variations in channel and pipe flows resulting from dissipation effects at high Mach numbers lead to changes in turbulence anisotropy. Coleman et al. [5] found that the near-wall streamwise elongated streaky structures in a channel flow become increasingly coherent at higher Mach numbers and showed that this is an effect of mean density variations. Figure 1 shows streamwise velocity fluctuations in subsonic and supersonic fully-developed pipe flow DNS in a streamwise-azimuthal plane (curved surface ‘rolled-out’ to a plane for visualization) in the near-wall region at y ∗ = 5. Here y ∗ = (R − r) ∗ u∗τ /¯ν is a wall-normal coordinate defined by Huang et al. [8] which takes into account the near-wall √ mean property variations, where ν¯ is the local kinematic viscosity and u∗τ = τw /ρ¯ defines a local friction velocity using the local mean density ρ. ¯ Red and blue contours indicate high-speed and low-speed regions, respectively. The increased length and thickness of the streaky structures at the supersonic Mach number is clearly visible in this plot. Figure 2 shows an instantaneous plot of the axial velocity fluctuations in the nozzle with incoming fully-developed supersonic pipe flow in a plane along the centerline and normal to the wall. The gradual acceleration effect in this nozzle is clearly visible in this picture. Near-wall sweep-ejection activity, which is the primary mechanism of turbulence production, reduces as we move downstream along the nozzle. The flow near the exit shows a reduced state of turbulence, but is not relaminarized.

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Fig. 1 Instantaneous axial velocity fluctuations (DNS) in fully-developed pipe flow in a (x, φ) plane for M = 0.3 (top) and M = 1.5 (bottom) at y ∗ = 5. Red and blue lines show positive and negative fluctuations respectively

In the following, we discuss mean flow features in the nozzle, the evolution of Reynolds stresses and Reynolds stress budgets with a focus on pressure-strain correlations. Most of the results are based on LES data. DNS data are occasionally used to validate the results. For an improved understanding of compressibility and acceleration effects it is necessary to discuss the behavior of mean primitive flow variables first. Figure 3 (left) shows axial profiles of mean centerline Mach number and pressure (normalized with its value at inflow). Figure 3 (right) shows profiles of mean temperature and density along the nozzle centerline. In both figures results obtained from isentropic streamtube equations have been plotted for comparison. The mean core flow behaves close to accelerated isentropic flow with decreasing pressure, density and


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Fig. 2 Instantaneous axial velocity fluctuations (DNS), normalized with uτ at x/L = 0, in a (x, r) plane of the nozzle

Fig. 3 Centerline distributions of mean pressure, local Mach number (left) and mean density, temperature (right). Solid line: LES, Dashed line: Isentropic streamtube equation

temperature. Figure 4 shows the axial evolution of radial temperature and density profiles and a comparison of DNS and LES data, which proves good agreement. In fully developed pipe flow (x/L = 0.0) mean density and temperature are directly linked in radial direction, since the radial pressure gradient is negligibly small. The heat generated by dissipation in the wall layer strongly increases the mean temperature and leads to a heat flux out of the pipe. The mean density in turn drops from its high wall value to a low core value and thus reduces the pressure-strain correlations. This effect, explained for fully-developed turbulent channel flow by Foysi et al. [6], also holds for fully-developed pipe flow. The described direct coupling between temperature and density in radial direction persists in the present accelerated flow (Fig. 4, x/L > 0.0). While in the nozzle core adiabatic cooling due to acceleration compensates dissipative heating and decreases T¯ /Tw , this effect is less pronounced in the near-wall region. The fact that the ratio ρ/ ¯ ρ¯w shows a downstream increase in this plot is counterintuitive and is due to the faster decrease of ρ¯w along the nozzle wall. Due to flow acceleration in the nozzle the mean sonic line moves closer to the wall, so that the layer in which subsonic flow persists gets thinner in downstream direction. The strong increase in wall shear stress due to acceleration and the weaker

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Fig. 4 Mean density, temperature profiles in the nozzle at stations x/L = 0.0(... ...), 0.45 (—), 0.8(−. − .−). Lines: LES, Symbols: DNS

Fig. 5 Van Driest transformed velocity. LES results only. x/L stations as in Fig. 4

increase in mean density ratio combine in such a way that the Van Driest transformed velocity profiles develop as shown in Fig. 5. Similar effects were observed by Bae et al. [2] in DNS of strongly heated air flow in pipes. Flow acceleration dramatically affects the turbulence structure. The axial Reynolds stress, normalized with the local wall shear stress, decreases by nearly an order of magnitude, as seen in Fig. 6. Due to non-equilibrium of the flow, τw is no longer a scaling parameter suitable for collapsing Reynolds stress profiles in the core region, as is the case of fully-developed pipe flow [7]. Figure 6 also evaluates the LES data by comparison with DNS data. The slight overshoot of the peak value in the LES is a consequence of ADM which does not account for the local anisotropy of the velocity fluctuations. Figure 7 presents the downstream evolution of the Reynolds shear stress and the total shear stress. Here again, a dramatic decrease of both quantities in the flow direction is observed. A similar strong decay of the solenoidal TKE dissipation rate and a decrease of the peak value of the turbulent Mach number from 0.25 to 0.17 is also noticed (not shown here).


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Fig. 6 Axial Reynolds stress in the nozzle normalized with local wall shear stress. x/L stations as in Fig. 4. Lines: LES, Symbols: DNS

Fig. 7 Reynolds shear stress and total shear stress in the nozzle normalized with local wall shear stress. LES results only. x/L stations as in Fig. 4

In order to understand the reasons for these changes in the nozzle, we examine production terms and pressure-strain correlations in the Reynolds stress budgets of ρux ux /2, ρux ur , ρur ur /2 and express them in a cylindrical (x, φ, r)-coordinate system which differs only weakly from the computational coordinate system. In such a system the radial budget contains production terms as well. We distinguish between ‘kinetic’ and enthalpic production and split the first into contributions due to shear, extra rate of strain and mean dilatation. The ‘kinetic’ production terms are: ∂ ux 1 ∂ ul ∂ ux 1 ∂ ul − − ρux ux −ρux ux , Pxx = −ρux ur ∂x 3 ∂xl ∂r 3 ∂xl shear

mean dilatation

extra rate of strain

∂ ux ∂ ux ∂ ur 2 ∂ ul ∂ ur 2 ∂ ul −ρux ux − ρux ur −ρux ur , + − ∂x ∂r 3 ∂xl ∂r ∂x 3 ∂xl

Pxr = −ρur ur

shear1

shear2

mean dilatation


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∂ ur ∂ ur 1 ∂ ul 1 ∂ ul Prr = −ρux ur − ρur ur −ρur ur . − ∂r 3 ∂xl ∂x 3 ∂xl shear

mean dilatation


Figures 8 and 9 show the contributions to Pxx and Pxr in the nozzle at x/L = 0.45 and in fully-developed pipe flow (x/L = 0), normalized with local values of τw2 /μ. ¯ In these and the remaining figures, only LES data are plotted for clarity. Among the two production by shear terms in Pxr , the first is dominant and the second is negligible in this specific nozzle. Clearly, compressibility in the form of mean dilatation counteracts the shear production of the ρux ux , ρux ur components. Acceleration (extra rate of strain) does the same, at least in the axial component. The production rates by shear are themselves reduced by the stabilization of the two stresses ρux ur and ρur ur . In the ρux ux -budget enthalpic production appears on the RHS in the form −ux ∂∂xp¯ and has only a very small positive value (not shown).

Fig. 8 Contributions to the production of the axial Reynolds stress in the nozzle at stations x/L = 0.0 (dashed line) and 0.45 (solid line). sh: mean shear, dil: mean dilatation, es: extra rate of strain

Fig. 9 Contributions to the production of the Reynolds shear stress in the nozzle at stations x/L = 0.0, 0.45. sh1: mean shear (shear1), dil: mean dilatation


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Fig. 10 Pressure-strain correlations: Πxx (left), Πrr (right) in the nozzle. Lines represent deviatoric parts and ‘◦’ stands for p d /3 at x/L = 0.45. x/L stations as in Fig. 4

The pressure-strain correlations can be split into deviatoric and dilatational parts:

Πxr Πrr

∂ux d − ∂x 3

1 + p d , 3 ∂ur 1 ∂rux + , = p ∂x r ∂r ∂rur d 1 1 − + p d = p r ∂r 3 3

Πxx =

p

d represents dilatational fluctuations. Profiles of Πxx , Πxr , normalized with local values of τw2 /μ, ¯ are presented in Fig. 10 for stations x/L = 0.0, 0.45, 0.8. The dramatic reduction of the deviatoric parts in the nozzle is obvious. The contribution of the pressure-dilatation correlation is very small. Besides the weak production by shear, the Πrr -term is the only source term in the radial stress budget. The axial decay of ρur ur is therefore mainly due to the reduction of the pressure-strain correlation which can be traced back to the reduction of pressure and velocity-gradient fluctuations. It remains to be shown which role mean dilatation, extra strain rate and mean density variation play in damping pressure fluctuations.

4 Conclusions Supersonic turbulent pipe flow subjected to gradual acceleration in a nozzle has been investigated by means of DNS and LES in order to assess the effects of mean dilatation and extra rate of strain on the turbulence structure. Although the rates of acceleration are small, the decrease in Reynolds stress components is large. At the same time dilatational fluctuations are only weakly affected, so that explicit compressibility terms (like pressure-dilatation and compressible dissipation rate) remain small.

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Among the source/sink terms in the Reynolds stress transport equations the production by shear, by extra-rate of strain, by mean dilatation and the pressure-strain correlations are strongly affected. It remains to be shown in which way extra-rates of strain, mean dilatation and mean density variations affect pressure and velocitygradient fluctuations and thus control the variation of pressure-strain correlations. This is the aim of future work. Acknowledgements The computations were performed on the SR8000 and the Altix 4700 at Leibnitz Rechenzentum. The constant support of the LRZ staff is gratefully acknowledged.

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