Fluid mechanics

Experiments on stability, transition, separation and turbulence in boundary layer flows

Researchers: Henrik Alfredsson, Per Elofsson, Carl Häggmark, Masaharu Matsubara, Nils Tillmark, Johan Westin

Sponsors: NUTEK, TFR, KVA, Göran Gustafsson stiftelse

The transition between laminar and turbulent flow is an important aspect of the design of e.g. turbo-machinery or vehicles, and effects for instance heat exchange and friction loss. From a technical point of view, it is therefore important to be able to predict the transition region and if possible also to control it. Fluid dynamic research has led to a fairly good understanding of laminar-turbulent transition in the absence of ambient disturbances and resulted in reliable transition prediction methods for flow on for instance aircraft wings where external (atmospheric) disturbances are small. Under less ideal conditions encountered for other technical applications, ambient disturbances or effects of body forces may cause earlier transition. However, in this case, the prediction of the transition relies so far upon uncertain empirical correlations, since the transition mechanisms are presently unknown.

The receptivity of the laminar boundary layer to free stream turbulence is investigated through detailed velocity measurements in the MTL wind tunnel, where free stream turbulence is generated by different grids. Both flow visualisation and hot-wire measurements (one and two-point) have shown that the interaction with the boundary layer gives rise to elongated structures of high and low velocity. These streaks seem to be susceptible to secondary instabilities and subsequent breakdown. This interaction is presently studied. This research is carried out partly in cooperation with scientists from the Institute of Applied and Theoretical Mechanics, Novosibirsk.

Formation of elongated structures may also occur through the interaction between two finite-amplitude oblique waves. This is experimentally investigated by introducing two oblique TS-waves in an air flow channel by means of vibrating ribbons (one at each channel wall). Measurements were carried out at Reynolds numbers of about one third of the critical one, and at an initial streamwise disturbance level of the order of 1-4 %. With only one ribbon excited the oblique wave decayed downstream in accordance with linear theory, but with the excitation of both ribbons dramatic changes of the flow field occurred. A strong mean flow distortion in the form of a spanwise peak-valley structure and a strong growth of higher harmonics prior to breakdown were observed. Also numerical simulations where the ribbons are modelled with volume forces have been made. Excellent agreement between experiments and simulations were obtained.

Boundary layer separation is an important aspect of flows over surfaces, e.g. it is well known that a large scale separation on for instance an airfoil at high angle of attack can give rise to large changes in lift and drag forces. In other situations separation may occur only in a limited region, on an airfoil for instance close to the leading edge where the suction peak may give an adverse pressure gradient strong enough to give a localised separation. Such a separation often induces transition to turbulence, which in turn usually causes reattachment. This project has just started and the aim is to improve the physical understanding of the process leading to transition in flows with separation. A related study of numerical simulations of separated flows is carried out by Prof. Henningson.

In cases where body forces affect boundary layer flows different other types of instability may be dominating. For instance a boundary layer flow along a wall in a rotating system will be affected by a Coriolis force which can give rise to instabilities in the form of longitudinal vortices. Similar vortices can develop along concave walls due to centrifugal effects. These instabilities may arise at much lower Reynolds numbers than the traditional TS-wave instability. The enhanced heat transfer of such vortices as well as secondary instability are studied through controlled experiments in an air channel with system rotation. Both temperature and velocity measurements are made with hot-wire technique.

Turbulent plane Couette flow is one of the canonical flow cases and among its interesting and distinguishing (as compared to pressure driven flows) features are the monotonous velocity profile, the constant shear stress distribution and the finite production rate of turbulent energy at all positions across the channel. The present apparatus uses an "infinite" plastic band moving along vertical glass surfaces in water, with spanwise and streamwise aspect ratios of the channel larger than 30 and 150, respectively. By using a transparent plastic band it is possible to use Laser-Doppler-Velocimetry for non-intrusive measurements and of special interest is two-point correlations of the streamwise velocity. In comparison to other wall bounded shear flows the distance over which streamwise correlation is found is exceptionally long (larger than 30h where h is the half-channel width). There is also a regular periodic structure in the spanwise direction with a wavelength of about . This indicates that there exist large scale structures in turbulent plane Couette flow that do not have a counterpart in for instance plane Poiseuille flow and that this probably leads to large difficulties in trying to simulate this flow. A cooperation with researchers from NTH-Trondheim and prof. Johansson's group carrying out DNS for this flow is on-going. Tillmark defended his doctoral thesis in May 1995 within this project.

Publications: 9,10,12,37,54,58,66,89,95,104,121,122

Boundary layer transition -- Theory and DNS

Researchers: Dan Henningson, Stellan Berlin, Ardeshir Hanifi, Casper Hildings, Anders Lundbladh

Sponsors: NUTEK,TFR,FFA

This project involves reserach to determine the maximum growh possible of disturbances evolving according to linear theory, as well as to investigate the importance of this growth when non-linearity comes into play. Several shear flow types have been considered. The results show that non-modal growth, i.e. growth not associated with individual eigenmodes but inherently dependent on their superposition, can cause large transient amplification. This growth is mainly associated with streaky structures in the streamwise direction. A new flow investigated recently is the compressible boundary layer. Large transient growth was found similar to the incompressible case. The growth was found to increase with Mach number. Detailed investigation of the non-linear energy transfer between Fourier components in plane Poiseuille flow shows that energy supplied to smaller scales during transition mainly originates from the non-modal transient growth. Non-linear calculations have shown that when the optimal disturbances from linear theory are used as initial conditions, the threshold amplitudes required for transition to turbulence is lower than for general disturbances.

Another part of the project involves direct numerical simulations (DNS) of transition to turbulence where these tranisent growth mechanisms play a major role. This bypass of the traditional Tollmien-Schlichting instability waves is involved in many shear flow transition scenarios. Previously transition associated with localized disturbances have been investigated, and at present two other bypass scenarios are under consideration. First, the transition in boundary layers starting with a pair of oblique waves is investigated. These waves generate elongated structures in the streamwise velocity which rapidly grow due to the non-modal mechanism. The resulting streaks break down to turbulence due to a secondary instability. Second, investigations of transition in boundary layers due to turbulence in the free-stream is simulated by introducing oblique disturbances in the free stream. These generate growing streaky structures inside the boundary layer, closely resembling those experimentally found in a laminar boundary subjected to free stream turbulence. Experiments have shown that the streaky structures grow algebraically, indicating that a similar non-modal instability is operating as in the breakdown of oblique waves. Transition due to turbulence in the free-stream is important in many technological applications, e.g. in turbines and jet engines.

The project also involves DNS of transition to turbulence in three-dimensional boundary layers. Spatial development of disturbances in Falkner-Skan-Cooke boundary layers are under consideration. These flows are important models of aeronautically related flows, such as those over swept wings. A new direction in this research is the consideration of transition to turbulence in flows with separation bubbles, of high importance for aircraft in high-lift situations.

Publications: 11,23,24,25,30,39,42,67,80,81,88,99,110,111

Modern stability prediction methods

Researchers: Ardeshir Hanifi, Paul Andersson, Dan Henningson, Henrik Alfredsson.

Sponsors: Swedish National Space Board, FFA, NUTEK

The project concerns a new transition prediction tool which is being developed in cooperation with DLR in Göttingen. The code uses the parabolized stability equations (PSE) and is so far based on the linearized equations. The method uses a wave ansatz with a slowly varying amplitude function and wave number, similar to the WKB method. In addition an auxillary condition is introduced which ensures uniqueness of the solution so that the traditional WKB expansion can be avoided. This method has proven to be efficient and to produce accurate stability results for complicated flows. It has been carefully checked against existing solutions and will be extended to handle non-linear interactions between wave components. In a second phase of the project the mathematical nature of the PSE-equations are studied. It is found that the step size restriction found in numerical implementation of the equations can be removed after a careful introduction of new terms in the equations making them mathematically well posed.

Applications motivating the development of this method is here the hypersonic transition research carried out within the Swedish participation of the "SÄNGER"-project and that of transonic cascade flow. However the PSE-technique will probably find use in many other applications in the future.

Publications: 57,93,107,108

Modelling of solidification in materials processing

Researchers: Gustav Amberg, Robert Tönhardt

Sponsors: TFR, KTH (rörlig resurs).

During solidification, for example in casting or welding, mushy zones consisting of dendritic crystals often form. The properties of a finished casting are determined by the size and morphology of the crystals, and is often strongly affected by convective heat and mass transfer during solidification. This project is concerned with mathematical models for solidification in processes such as welding and near net shape casting. One part of this is to predict microstructure, i.e. the crystal structure and the size, geometry and orientation of crystals. The models developed within the project are to be incorporated in available codes for simulating the macroscopic convective heat and mass transfer during solidification. Development of mathematical models will require simulations of individual dendrites. Models and predictions will be continuously tested against experiments (in collaboration with Hasse Fredriksson, KTH).

Another issue which is studied is the rather complex dynamics of convective flow through the mushy layer, giving rise to well known defects such as macrosegregation and freckles. Such specific phenomena has been studied within this project and will be investigated further, using the code and models that are developed. These tools will also be used to simulate specific solidification processes in related projects.

Thermocapillary convection in materials processing.

Researchers: Gustav Amberg, Henrik Alfredsson, Christian Winkler

Sponsors: NUTEK, TFR, KTH (rörlig resurs)

If surface tension depends on temperature, a fluid motion will be induced along a free surface with a temperature gradient. This is an important phenomenon in many materials processes, characterized by large temperature gradients, small volumes of liquid metal, and the presence of free surfaces. This convection is often crucial for the properties of the finished product. Examples of such processes are all the various techniques for crystal growth, and welding, where the flow in the weld pool determines the penetration of the liquid pool (i.e. 'weldability'). Often it is technically important to avoid oscillatory flow, and thus it is important to understand the stability characteristics of thermocapillary convection in general.

A preliminary experimental study of the transition from stationary to oscillatory motion in buoyant thermocapillary convection has been made. Instability was observed through flow visualization in qualitative agreement with numerical calculations. Future experiments are aimed at obtaining alos quantitive results for the stability boundaries of such flows. The emphasis will be on identifying instability mechanisms, in conjunction with numerical simulations.

Welding of the light metals Aluminum and Titanium today presents a number of practical difficulties. The flow in the melt during welding of Al and Ti alloys will be studied by numerical simulation, using tools and models developed in accompanying projects. This will be closely coupled to an experimental study of Al and Ti welding carried out by Torbjörn Carlberg, Sundsvall.

Another process where thermocapillary convection is crucial is float zone crystal growth. The stability of the flow in such processes are simulated numerically and comparisons are made with actual float zone experiments in space and on earth (Torbjörn Carlberg, Sundsvall).

(Cooperation with Torbjörn Carlberg, Sundsvall, and Mårten Levenstam, CTH)

Publications: 34,98,115,117

Convection in electrochemical systems

Researchers: Fritz Bark, Rolf Karlsson, Daniel Simonsson*, Rikard Eriksson*, Lars-Göran Sundström, Fredrik Wallgren

Sponsors: NUTEK, TFR

In many electrochemical systems, transport of ionic species takes place due to migration in the electric field, diffusion due to concentration gradients and advection by a moving liquid electrolyte. Examples of such systems are lead acid batteries and cells for refining of metals. The purpose of the present project, which involves both theoretical modeling and experimental studies, is to provide the electrochemical industry with theoretical design tools and experimental methods for validation. The theoretical modeling involves approximate analytic methods and numerical analysis. In the experimental studies, concentration and velocity fields are measured by using laser holography and laser Doppler velocimetry, respectively. Attention is focussed on the nonlinear coupling between the electrode kinetics and the aforementioned transport mechanisms. Also the formation of dendrites and some fundamental electrokinetic phenomena are investigated theoretically. The studies have so far been restricted to continuous phases. In a longer perspective, however, effects of gas bubbles in the electrolyte will be considered. This project is a joint venture with the Department of Chemical Engineering and Technology, KTH.
* Affiliated with the Department of Chemical Engineering and Technology, KTH.

Publications: 7,8,28,29,43,44,45,47,62,114,119,123,124

Fluid mechanics of twin wire paper machines

Researchers: Fritz Bark, Rolf Karlsson, Bo Norman*, Nicholas Moch, Mats Nigam, Sima Zahrai

Sponsors: NUTEK, Bo Rydin Foundation, Swedish Forestry Research Foundation, Swedish Pulp and Paper Research Institute (STFI)

In modern paper machines, the wood fibre suspension is de-watered through two permeable webs. These webs, which are called wires, are highly prestressed. A significant part of the water content is removed by having the pair of wires run over a permeable roll. After the passage over the roll, the direction of motion of the wires is deflected slightly a number of times (in alternating directions) by moving the wires over a set of blades. The motion over the blades sets up a straining field in the suspension, which probably causes break up of fibre flocs. This is a highly desirable process as the paper quality is thereby improved. The wire deflection also leads to some further de-watering. The purpose of the project is to provide the paper industry with a quantitative model of the procceses taking place during the passage of the wires over the blades. With reasonable choices of so far not measured parameter values, theoretical predictions agree quite well with measurements of pressure on blades carried out at STFI.
* Affiliated with the Department of paper and pulp technology, KTH.

Publications: 61

Fluid dynamics of plane liquid jets

Researchers: Henrik Alfredsson, Daniel Söderberg

Sponsors: NUTEK

This project deals with the stability and break-up of liquid jets, emanating from a plane nozzle and is motivated by the paper making industry where plane jets of a low concentration fibre suspension distributes the fibres to the paper machine. For a typical paper machine the jet width can be 10 m with a thickness of about 1 cm, and velocities of the order 20-30 m/s. The flow is ideally two-dimensional, however, paper is usually not perfectly homogenous across its width, showing that the jet flow is not perfect. This can be due to inhomogeneities in the jet contraction, centrifugal instability in the form of Dean vortices or inhomogenous break-up of the jet. For plane jets the surface is affected both by a surface tension force which always tends to restore the interface back to its original equilibrium position, whereas the aerodynamic forces developing at the interface between liquid and gas enhance the instability. This may cause the instability to grow until the liquid sheet disintegrates and splits up into droplets. The research program aims at increasing our knowledge about the development of two-dimensional liquid jets in air, both for Newtonian liquids (i.e. pure water) and fibre suspensions typical for paper making.

Publications: 120

Sedimenting two-phase flow

Researchers: Anders Dahlkild

Sponsors: 'Rörlig resurs', KTH, Göran Gustafssons Stiftelse

Problems of centrifugal and gravitational sedimentation of small particles suspended in a fluid are studied theoretically. A fundamental problem in separation processes is the removal of accumulated material of settled particles on the container walls. The models of shear induced resuspension of particles has been complemented to account also for Brownian diffusion of particles. In particular, the accumulation and flow of particles down an incline are studied. With Brownian diffusion included steady solutions are found in parameter regimes which otherwise would lead to a continuous packing of a very dense sediment. Additional extensions of the model includes a yield stress due to interparticle friction appearing at elevated concentrations. Continued work will investigate these models for a sediment in a rotating environment for which novel effects can be expected. Existing numerical schemes will also be extended to a scheme for bidisperse suspensions, which is largely an unexplored area for rotating systems.

Publications: 13,101

Theoretical studies of the wave-number spectra of turbulence

Researchers: Erik Lindborg, Anthony Burden, Arne Johansson

Sponsors: TFR

This activity consisits of theoretical and numerical analysis of the statistical formulation of turbulence in wave-number space. Careful study of the invariants of the 2nd-order velocity correlation of axisymmetric, reflectionally symmetric, turbulence has demonstrated the feasability of evaluating the 2nd-order spectrum from measurements of only two velocity components along a single line orthogonal to the axis of symmetry. In principle, the pressure-strain-rate correlation and anisotropic dissipation rate can also be evaluated. Erik Lindborg has also considered the effects of system rotation, in particular on the -equation. An ongoing project is the study of the equilibrium range of wave numbers and in particular the Eddy-Damped Quasi-Normal closure procedure. The theoretical analysis is supported by direct simulation of Navier-Stokes turbulence at low Reynolds numbers as well as by numerical calculations using shell models at high Reynolds numbers.

Publications: 36

Measurement, modelling and simulation of turbulence

Researchers: A.V. Johansson, M. Hallbäck, T. Sjögren, P. Wikström, K. Alvelius

Sponsors: TFR, NUTEK

The aim of the project is to develop and critically evaluate models for statistical description of turbulent flows. The main methods used within the present project for gaining further knowledge of the physics of turbulence are experimental studies and direct numerical simulations of turbulent flows. These methods are complemented by so called rapid distortion analysis and to some extent also by spectral theories. The models so far investigated belong to the realm of one-point closures for the turbulent stress tensor and turbulent heat flux vector to be used for computational fluid dynamics. The main emphasis is laid on closures based on the transport equations for the turbulent stresses. Particularly, our efforts have been focused on the modeling problems of flows exhibiting strongly anisotropic turbulence. A number of terms responsible for intercomponent energy redistribution have been scrutinized and models for the individual terms have been proposed. The models are cast in continuum mechanics type of tensor formulations and satisfy basic principles such as realizability of the solutions produced by the closure.

The major part of the work on modelling within the present project has been limited to homogeneous turbulence. Investigation of near-wall and other inhomogeneity effects have, though, been initiated and some preliminary results from direct simulations of a so called free (stress-less) surface flow have been obtained. The most important result is that the pressure transport is an important term in the near wall anisotropic region.

Detailed measurements using arrangements of double X hot-wires have been performed of the individual components of the Reynolds stress tensor and the two-point second and third order velocity correlations. From these measurements information have been extracted concerning the intercomponent energy transfer terms. The results agree well with earlier numerical simulation results and a proposed model for the Reynolds-number dependence of the so called slow pressure-strain-rate correlation.

Work on turbulence in a rotating system focusing on the energy cascade mechanism has been pursued. The perhaps most striking feature of turbulence in a rotating frame is the decreased rate of turbulence kinetic energy decay (see figure below). Rapid distortion analysis was used to determine the influence on the energy cascade intensity from imposed system rotation. A single point closure was proposed that accounts for the decreased cascade intensity and for the imbalance between the energy cascade rate and the viscous dissipation rate during a transient portion, after the ``turn on'' of system rotation, of the lapse of decay. Usually turbulence models rely on an assumed equality between these two energy-rate-of-change processes.

The research on turbulence modelling for passive scalars has been continued. The working process is analogous to that of the turbulent stress modelling research, involving experimental work and single-point closure formulations. So far the main effort has been directed to experimental work, among other things involving development of constant current anemometry probes for fluctuating scalar derivative measurements.

During the past year work on large-eddy simulation and sub-grid-scale modelling has been initiated. A number of sub-grid-scale models have been implemented in a spectral simulation code. Both LES-predictions and a priori comparisons with DNS-data have been made to investigate the properties of a variety of proposed sub-grid-scale models.

Publications: 26,27,82,94

Numerical simulation of fully developed turbulent plane Couette flow

Researchers: Arne Johansson, Jukka Komminaho, Anders Lundbladh (FFA)

Sponsors: NUTEK

Fully developed turbulence in plane Couette flow is studied by means of direct numerical simulation. Experience has shown that this is a particularly difficult case to study because of a tendency to develop extremely long vortical structures aligned in the streamwise direction. For a numerical simulation study an extremely long (and also rather wide) box is needed, almost 90 half-heights long in the present case. The computations were carried out on a massively parallel computer (CM200) at KTH where the calculations amounted to a total of about one CPU-month. Accurate statistics have been acquired and the long structures have been studied in detail. It was also shown that a weak (spanwise) rotation has a drastic effect on the long structures. Comparisons with experimental results of Tillmark & Alfredsson show excellent agreement. Also the relaminarization of plane Couette flow turbulence was studied by a step-wise lowering of the Reynolds number (see figure). This was achieved by increasing the viscosity, thereby avoiding the instabilities that would occur if the velocity suddenly was decreased. The transition Reynolds number could hereby be determined to about 360 thereby substantiating the previous numerical/experimental findings of Lundbladh & Johansson and Tillmark & Alfredsson. Also the turbulence regeneration mechanisms could be elucidated at low Reynolds numbers. Strong similarities were found with the streak instability mechanisms observed in typical bypass transition scenarious in various flows.

Publications: 112,113

Turbulent boundary layers at high Reynolds numbers

Researchers: Arne Johansson, Jens Österlund

Sponsors: NUTEK

For turbulent boundary layers typical Reynolds numbers are in most applications very high, whereas most laboratory experiments have been carried out at low to moderate Re. In the present project boundary layer measurements are carried out in the MTL wind tunnel at KTH, on a 7 m long boundary layer plate and with free-stream velocities up to 50 m/s. This gives Reynolds numbers based on momentum loss thickness of up to 20000 or roughly 20 million based on x, which is realistic for practical applications. Hot-wire anemometry is used with X-probes with box sides down to 0.10 mm. However, severe restrictions in the method have been identified that are coupled to interaction of the thermal wakes from the wires (occurring at low Peclet numbers). Also new types of probe geometries are tested. The thermal wake interactions have been studied in detail by means (2D) numerical computations of heat transfer from two point sources subjected to a fluctuating velocity field. Several conclusions have been reached from the numerical results regarding the problems in different parameter regimes and on criteria for suitable design of two-wire probes. A new traversing equipment especially suited for measurements in the near-wall region has been constructed and new measurements using single and double probe arrangements are under way.

Publications: 97

Development of 3D LDV measurement techniques with applications to wall bounded shear flows

Researchers: Rolf Karlsson, Jan Eriksson

Sponsors: NUTEK, Vattenfall Utveckling AB

The aim of the project is to develop a practically useful methodology for making simultaneous 3D LDV measurements with high spatial and temporal resolution, and to apply this technique to obtain detailed 3D turbulence data in the plane turbulent wall jet. In a longer perspective, such data will be used to improve near-wall Reynolds stress turbulence modelling. The first phase of the project has now been successfully completed, and measurements in an enclosed circular jet with a measuring volume as small as have been made.

The second phase of the project is to supplement an earlier (2D) experimental investigation of the turbulent wall jet with simultaneous 3D measurements of the total velocity vector. In particular, attention will be focussed on the equation for the turbulent kinetic energy and on the limiting behaviour of the Reynolds stresses near the wall. Such measurements are presently going on.

Publications: 18,73

Hypersonic afterbody flow fields

Researchers: Tor-Arne Grönland, Anders Dahlkild

Sponsors: ESA

This work is part of a research project performed external to KTH by the team FFA and DASA. The aim of the study is to make a thorough and basic investigation of the importance of different physical and geometrical effects which influence the efficiency and versatility of a hypersonic afterbody design. The complete propulsion system is an integrated part of the airframe of a hypersonic airbreathing vehicle. The vehicle body will act as expansion surface, yielding an unsymmetric expansion of the engine exhaust gases to the surrounding pressure. In the design of such an afterbody there are a number of critical issues of which one needs a thorough knowledge.

Publications: 75

Large Eddy Simulation of swirling jets

Researchers: Magnus Olsson, Laszlo Fuchs

Sponsor: TFR

"Dynamic" Large-Eddy-Simulations (D-LES) have the advantage that the model is parameter free. It also offers the possibility of treating transitional flows as the model "shuts" itself "down" in laminar regions. One of the main issues that is addressed here is the way of determining the validity of the basic assumption that the model has reached an "asymptotic" behaviour. Furthermore, one has to be able to distinguish between the truncation errors and the generalized moments (which may contain numerical errors due to the way the double spatial filtering is carried out!). The technique is tested/applied to spatially developing jets and to the mixing (of passive scalar) in annular swirling jets. The computations are compared to measured data (from LTH).

Publications: 91,92,116

Reacting flow modelling

Researchers: Xue-Song Bai, Laszlo Fuchs

Sponsor: NUTEK

Models for the interaction between turbulence and chemical reactions are studied and analyzed. These models include EDC (Eddy-Dissipation Concept), PDF and flamelet models. The models include some adaptivity in terms of determining which chemical reactions (of a reduced set of reactions) have to be computed together with the flow solver and which reactions are slow (low Damköhler number) and thus can be computed as part of a "post-processing". The sensitivity of different models to the numerical value of model paramters has been studied. These type of studies indicate when and for what purposes, one may use a particular model. Simulation of reacting flows in furnaces and gasturbine combustion chambers have been carried out. The combined models have also been used for geometrical optimization of a specific furnace. In these calculations even radiative heat-transfer was taken into account.

Publications: 4,63,64,65

Numerical methods for compressible flows

Researchers: Pär Ekstrand, Laszlo Fuchs

Sponsor: NUTEK/EU (BRITE-EURAM)

For efficiently solving 3D transonic flows, a Multi-grid based, Finite-volume Multi-block/overlapping-grid code has been developed. Within this BRITE-EURAM-II project we are developing methods for generating adaptive grids to be used in conjunction with fast Multi-grid solvers, in low and high Mach numbers flows. The method includes a mesh adaptive techniqe for improving resolution when required. Also, a solution reconstruction technique allows improved accuracy in flows with shocks, without requiring high level of local grid refinement.

Publications: 68,74