Large Eddy Simulation of a Pseudo-shock System Within a Laval Nozzle

Principal Investigator:
Christian Stemmer, Stefan Hickel

Technische Universität München, Fakultät für Maschinenwesen

Local Project ID:

HPC Platform used:
SuperMUC and SuperMUC-NG of LRZ

Date published:

Introduction and Motivation

The flow conditions in Laval nozzles (convergent-divergent nozzle geometry) are defined by the pressure ratio between inlet and outlet. If this ratio is sufficiently high, the flow is accelerated in the convergent section until sonic speed (Mach number M = 1) is reached at the location with the smallest cross section. In the subsequent divergent part of the nozzle the flow is further accelerated to supersonic speed. This acceleration leads to a decrease of the static pressure. If the outlet pressure is higher than the pressure defined by the nozzle geometry, a shock occurs in the divergent part of the Laval nozzle, which instantaneously decelerates the flow to subsonic conditions and ensures compliance with the pressure imposed at the outlet.

In the quasi-one-dimensional inviscid theory, the flow is decelerated by a single normal shock that leads to a jump of static pressure, temperature and density. Taking viscous effects and the three-dimensional channel geometry into account, the shock interacts with the turbulent boundary layers at the channel walls and causes flow separation, which is a particularly challenging subject of enduring fluid dynamics research. Flow separation and reattachment result in a complex three-dimensional system of interacting oblique shocks, compression waves and expansion waves. These so-called pseudo-shock systems appear in a wide variety of flow devices – ranging from ducts and pipelines in the field of process engineering to supersonic aircraft inlets - and influence reliability and performance. Thus, the comprehension and control of pseudo-shock systems are of great academic and commercial interest.

The purpose of this project is the numerical investigation of a pseudo-shock system. Prior Reynolds-averaged Navier-Stokes (RANS) simulations showed that the results are very sensitive on the applied modeling parameters (Giglmaier et al., 2014; Quaatz et al., 2014). For this reason, highly accurate Large Eddy Simulations (LES) have been performed during this project.

Setup and Numerical Method

The numerical simulations exactly reproduce the geometry and operating condition (Mach number, Reynolds number, pressure ratio) of experiments conducted at the DLR in Cologne by Gawehn et al. (2010).  Large Eddy Simulations of such a complex and fully turbulent (Reδ≈105) flow incorporating the real geometry and operating conditions require an adequate spatial and temporal resolution. In the present case this resulted in approx. 500 million finite volumes and a time-step size of about 2 ns. With five solution variables for compressible flows this amounts to more than 1∙1015 unknown degrees of freedom per millisecond of physical simulation time. Hence, the necessary computational power can only be provided by modern supercomputers and appropriate software.

The presented simulations were performed with the in-house multi-physics flow solver INCA (Hickel et al., 2014, Örley et al., 2015) on several thousand CPUs in parallel at LRZ SuperMUC Phase 1 and 2, and consumed more than 40 Mio CPUh. In return, a very detailed insight into the inner structure and the physical mechanisms within the flow field is achieved. This provides an excellent basis for foundational research into the flow physics of such pseudo-shock systems, for turbulence research and for the validation of alternative and simplified solution approaches.

Results and Achievements

The numerical results of our LES are validated against experimental data (Gawehn et al., 2010) for the same operating conditions. These experiments provide wall-pressure measurements as well as schlieren pictures (explained below). Figure 1 shows the experimental pressure along the upper channel wall (symbols) and the results of the corresponding LES (solid line) in the region of the pseudo-shock system. The flow is from left to right. Within the supersonic part a decrease in pressure due to the further acceleration of the flow can be observed. The onset position of the pseudo-shock system is characterized by a sudden rise of the wall pressure followed by a successive compression due to the deceleration of the flow to subsonic conditions. All these features are captured very well by the numerical LES results including the match of the onset of the pseudo-shock system.

Schlieren pictures as presented in Figure 2 visualize the density gradients within the flow by optical measurements. For this experimental set-up, the axial density gradient is visualized and compressions such as shocks appear black while expansions are bright. The advantage in contrast to wall measurements is that not only the global position of the pseudo-shock can be observed but moreover an impression of the structure is obtained. It becomes obvious that the pseudo-shock system consists not only of a single normal shock as proposed by the ideal inviscid theory. Rather, interactions with the boundary layers at the channel walls lead to a complex system of several shock events.

The first one shows a bifurcated shape: Oblique separation shocks starting in the near-wall region interact near the centerline where very small normal shock occurs. It is followed by several curved shocks which have an increasing distance to the wall. This behavior is reproduced very well by the LES (Figure 2). The number as well as the shape and axial distance of the shock waves are in good agreement with the experimental data by Gawehn et al. (2010).

Figure 3 gives an impression of the three-dimensional structure based on the LES results and gives detailed insights into the inner structure and the flow conditions of the pseudo-shock system which cannot be gained by experiments at all. The yellow iso-surface visualizes the border between super- and subsonic regions of the time averaged flow field (defined by Mach number M=1). Due to the channel geometry, the extend of the normal part of the bifurcated initial shock is bigger in channel depth than in channel height. Due to the consecutive total pressure loss the supersonic region becomes smaller for every single shock of the series. This is accompanied by a decreasing extent of the shocks in the schlieren pictures. The blue iso-surfaces enclose backflow regions and therefore characterize flow separations which bend the supersonic flow towards the channel center. Finally, only weak supersonic parts remain in a torus-like structure around the channel center. This part of the pseudo-shock system is called mixing region because deceleration is no longer achieved by shocks but solely by mixing of momentum between super- and subsonic flow regions.

Although the mean position of the pseudo-shock system within the Laval nozzle of the experimental facility is well defined by the boundary conditions, it is oscillating around that position and the overall structure is highly transient. Moreover, the performance and even reliability of facilities incorporating pseudo-shock systems are distinctly affected when the shock-train is bent to one of the diverging walls. During the presented simulation such a process occurred as depicted in Figure 4, where the two snapshots show the shock-train in centered position first (top) and later bent to the lower channel wall (bottom). The up and down bending of the pseudo shock system is a very slow process, much slower than the streamwise oscillations, and a very long integration time is needed to capture it in a simulation. Interpretation and understanding of this mechanism are of particular interest and can be supported by LES results.

Finally, due to the instant temperature rise pseudo-shock systems in Laval nozzles can be used for rapid heating of reactive flows in process engineering. Therefore, it is of crucial importance for the design of the facility how injected fuels are mixed with the ambient air flow. This mixing behavior has been investigated by adding a passive scalar to the LES. Figure 5 shows the propagation of a fuel injected at the centerline of the channel passing the shock-train (red surface in the supersonic and blue in the subsonic regions). Within the final mixing region of the pseudo-shock system the concentration of the fuel is spread over the channel cross section providing a good mixture.

Large Eddy Simulations provide a very detailed insight into the inner structure and the physical mechanisms within the flow field. This is an excellent basis for foundational research into the flow physics of such pseudo-shock systems, for turbulence research and for the validation of alternative and simplified solution approaches.

For the design of industrial facilities or devices it is important to have faster simplified tools, typically RANS simulations, that approximate the flow field with adequate accuracy at significantly lower cost than LES. Suitable candidates for model form and model parameters can be determined by comparison to the comprehensive LES results. The predicted shock structure highly depends on the choice of the RANS turbulence model as shown exemplarily for two common models in Figure 6. The LES clearly supports the results of omega-based Reynolds stress models represented here by the EARSM; results for other models can be found in Quaatz et al. (2014).


The simulations have been performed with INCA ( using approx. 4100 CPUs of the LRZ SuperMUC Phase 1 and 2 systems.


Gawehn, T., Gülhan, A., Giglmaier, M., Al-Hasan, N.S., Quaatz, J.F., Adams, N.A. (2010) Analysis of pseudo-shock system structure and asymmetry in Laval nozzles with parallel side walls, in: 19th International Shock Interaction Symposium, Moscow, Russia.

Giglmaier, M., Quaatz, J.F., Gawehn, T. Gülhan, T., Adams, N.A. (2014) Numerical and experimental investigations of pseudo-shock systems in a planar nozzle: impact of bypass mass flow due to narrow gaps. Shock Waves 24: 139-156.

Hickel, S., Egerer, C.P., Larsson, J. (2014) Subgrid-scale modeling for implicit Large Eddy Simulation of compressible flows and shock turbulence interaction. Physics of Fluids 26, 106101.    

Örley, F., Pasquariello, V., Hickel, S., Adams, N.A. (2015) Cut-element based immersed boundary method for moving geometries in compressible liquid flows with cavitation. Journal of Computational Physics 283: 1-22.

Quaatz, J.F., Giglmaier M., Hickel S., Adams N.A. (2014) Large-Eddy Simulation of a Pseudo-Shock System in a Laval Nozzle, International Journal of Heat and Fluid Flow 49: 108 – 115.


Project Team

N.A. Adams, M. Giglmaier, S. Hickel, and J.F. Quaatz
all: Technische Universität München, Fakultät für Maschinenwesen

Scientific Contacts

Dr.-Ing. Marcus Giglmaier
e-mail: marcus.giglmaier [@]

Dr.-Ing. Steffen Schmidt
e-mail: [@]

Apl. Prof. Dr.-Ing. habil. Christian Stemmer
e-mail: Christian.Stemmer [@]

Chair of Aerodynamics and Fluid Mechanics
Technical University of Munich
Boltzmannstr. 15, D-85748 Garching bei München (Germany)

NOTE: This report was first published in the book "High Performance Computing in Science and Engineering – Garching/Munich 2020 (2021)" (ISBN 978-3-9816675-4-7)

Local project ID: pr45tu

December 2021