Direct Numerical Simulation of a Spatially Developing Mixing Layer With Temperature Gradient
Institut Aérotechnique, Conservatoire National des Arts et Métiers, Saint-Cyr-l'Ecole (France)
Local Project ID:
HPC Platform used:
JUQUEEN of JSC
Direct numerical simulation (DNS) of turbulent mixing layers has been possible only in relatively recent times. In an ambitious project using HPC system JUQUEEN of JSC, scientists analysed the process of mixing layer formation in a flow configuration with sizeable compressibility effects by numerically reproducing the flow conditions of a well documented flow case with two turbulent streams. Large-scale direct numerical simulation were performed in a wide computational domain which included the two upstream turbulent boundary layers developing on the two sides of a zero-thickness splitter plate and their early merging region. The complexity of the flow, the extent of the computational box and the mesh size made the study extremely challenging in terms of CPU hours and memory requirements.
Turbulent mixing phenomena are ubiquitous in nature, from small-scale mixing in the bloodstream to the large scales of oceanic and atmospheric streams. This research subject has been extensively studied in the past and is also of major interest for industrial engineering systems, in which strongly heterogeneous flows can mix under a wide variety of schemes. Turbulence is known to be a major mixing catalyst, and its dynamical behaviour is a crucial issue for the understanding of mixing processes. Flow momentum redistribution is also responsible for the radiation of acoustic waves, and the study of mixing layers is a key building block for the understanding of jet-radiated noise.
Turbulent mixing is typically realized by merging of streams having different properties in terms of speed, temperature, and chemical composition, past a separating surface (in experiments, a splitter plate). Since early studies, it is known that large coherent structures form past the trailing edge of the splitter plate which are mainly organized as rollers with significant spanwise coherence, and being the main cause for the entrainment of fresh fluid into the mixing zone. Later, it became clear that the spatial organization of the coherent structures in the mixing layer is significantly affected by compressibility effects, by the state of the upstream boundary layers, and by the shape of the trailing edge of the splitter plate.
Regarding the first issue, the growth rate of the mixing layer is significantly reduced in the presence of mild compressibility effects, and the controlling parameter is the convective Mach number. Regarding the second issue, it should be noted that the state of the two merging streams may be laminar, turbulent, or transitional. In this respect, significant history effects, with influence of the upstream state even on the asymptotic structure of the developing mixing layer. Specifically, tripped upstream boundary layers yield lower growth rates than untripped boundary layers, although the mixing layer structure at about 150 boundary layer thicknesses downstream of the splitter plate is similar to within 10%, for the Reynolds stress components.
In most analyses presented in the literature both boundary layers are laminar, or one is laminar and one is turbulent. Although the (alleged) asymptotic developed state of turbulent mixing layers has been extensively studied, much less information is available from experiments regarding the non-similar region right past the trailing edge of the splitter plate, mainly because of technical difficulties in obtaining the necessary resolution in experimental facilities. In one of the few dedicated studies, to investigate the evolution of the wake forming past an airfoil trailing edge under low-speed conditions, counter-hairpin vortices form, as a result of the generation of spanwise vortices along the wake centreline, upon disappearance of the solid wall.
Direct numerical simulation (DNS) of turbulent mixing layers has been possible only in relatively recent times. To keep the computational cost within acceptable bounds, most numerical studies have been carried out in a temporal setting, by monitoring the time evolution of spatially periodic mixing layers. Temporal simulation have also shown that it is possible that the ultimate self-similar state may be dependent on details of the initial conditions, and provided insight into the growth reduction mechanism associated with compressibility effects.
Spatial mixing layer simulations have been rather limited, so far, and they are typically carried out by assuming a laminar mixing layer profile at the inflow with superposed small perturbations, consisting of a combination of the most unstable modes to accelerate transition to a turbulent state. This approach allows to capture some peculiar features of the spatial shear layer, including the non-symmetric spreading, but it suffers from the same limitations as temporal studies, in terms of unrealistic initial development. Notable exceptions include a study on the effect of trailing edge shape on the process of mixing, and the mixing of a laminar and a turbulent stream in the low-Mc regime by large eddy simulation (LES) including a splitter plate with zero thickness in the computation. These studies showed significant differences with respect to the case of mixing layers initiated through synthetic disturbances, and highlighted significant influence of the initial velocity deficit on the fully developed mixing layer region. In addition, a faster development of self-similar velocity profiles as compared to the case of two incoming laminar boundary layers is observed.
The goal of the present study is to analyse the process of mixing layer formation in a flow configuration with sizeable compressibility effects, by numerically reproducing the flow conditions of a well documented flow case with two turbulent streams. For that purpose we perform a large-scale direct numerical simulation in a wide computational domain which includes the two upstream turbulent boundary layers developing on the two sides of a zero- thickness splitter plate, and their early merging region. To our knowledge, this type of computation has not been attempted before. The complexity of the flow, the extent of the computational box and the mesh size make the study extremely challenging in terms of CPU hours and memory requirements. The HPC resources provided by the Jülich Supercomputing Centre, allocated through the Partnership for Advanced Computing in Europe (PRACE), have been the only viable way to achieve this ambitious goal.
The study has highlighted the predictive capabilities of DNS, in that the mean velocity profile and the main velocity statistics agree very well with reference experimental results. The flow is found to be organized into three parts: i) a near field (extending up to about 2-3 boundary layer thicknesses past the end of the splitter plate), which is substantially affected by the momentum deficit inherited from the parent boundary layers, and in which the Reynolds stress components attain strong peaks; ii) an intermediate field (extending up to about 20 boundary layer thickness past the splitter plate), in which the momentum deficit decays according to an inverse-square-root law, as in plane turbulent wakes, and iii) a far field in which momentum deficit is re-absorbed, and the flow converges to the classical mixing layer structure.
The analysis shows that the mixing layer growth in region i) is quite fast, occurring according to a power law of the trailing edge distance sensibly greater than unity. This observation (common to previous spatial studies) is interpreted as the consequence of incomplete entrainment of the boundary layers into the developing mixing layer. A qualitative change in the mixing layer growth takes place in regions ii) and iii), in which we observe slower, sub-linear growth. Correspondingly, we find decay of the Reynolds stress components according to a weakly negative powerlaw, without evidence of saturation as expected in fully self-similar mixing layers. This observation has been found to be the likely result of incomplete similarity, whereby mean momentum attains an equilibrium state prior to the turbulent stress distribution.
The lack of full self-similarity in the present study would then be the consequence of the limited streamwise extent of the computational domain. Establishing the spatial development scale for the achievement of full self-similarity would then be an interesting topic for a follow-up study. In terms of flow structures, we find very little hint of spatial organization in region i), whereas spanwise coherence is observed in region ii), where the typical eddies appear to be skewed with respect to the flow direction, and in region iii), where they tend to be aligned along the spanwise direction.
Notably, no distinct roller of the type found by Brown & Roshko has been observed, which is either due to compressibility effects, or again to limitation in the computational domain, or both.
As expected, the size of the velocity- and pressure-containing eddies is found to grow in the streamwise direction, apparently in approximately proportional fashion to the local vorticity thickness. In terms of the latter, the normalized spanwise integral scales are found to have values comparable to those of boundary layers (scaled by the boundary layer thickness), pointing to a universal behavior of the eddy size in all shear flows. The findings on the flow organization are well put into light by the dynamic mode decomposition (DMD) of the flow, which usefully isolates the process of mixing layer formation and saturation. In particular, DMD seems to provide hints on the formation of rollers, which are not (yet) clearly visible in the instantaneous flow realizations.
Project Team and Scientific Contact:
Matteo Bernardini, Assistant Professor
Ph.D. Francesco Grasso, Professor, Ph.D. (PI)
Simon Marié, Assistant Professor
Ph.D. Sergio Pirozzoli, Professor, Ph.D.
Pr. Francesco Grasso
Conservatoire National des Arts et Métiers
Titulaire de la chaire d'aérodynamique industrielle
Directeur de l'Institut Aérotechnique
15, rue Marat
78210 Saint-Cyr-l'Ecole (France)