Asymptotic theory of a uniform flow of a rarefied gas past a sphere at low Mach numbers

2015 ◽  
Vol 774 ◽  
pp. 363-394 ◽  
Author(s):  
Satoshi Taguchi
Keyword(s):  
Boltzmann Equation ◽  
Mach Number ◽  
Rarefied Gas ◽  
Fluid Dynamic ◽  
Uniform Flow ◽  
Second Order ◽  
Bgk Model ◽  
Weakly Nonlinear ◽  
Dynamic Type ◽  
Mach Numbers

A slow uniform flow of a rarefied gas past a sphere with a uniform temperature is considered. The steady behaviour of the gas is investigated on the basis of the Boltzmann equation by a systematic asymptotic analysis for small Mach numbers in the case where the Knudsen number is finite. Introducing a slowly varying solution whose length scale of variation is much larger than the sphere dimension, the fluid-dynamic-type equations describing the overall behaviour of the gas in the far region are derived. Then, the solution in the near region which varies on the scale of the sphere size, described by the linearised Boltzmann equation, and the solution in the far region, described by the fluid-dynamic-type equations, are sought in the form of a Mach number expansion up to the second order, in a way that they are joined in the intermediate overlapping region. As a result, the drag is derived up to the second order of the Mach number, which formally extends the linear drag obtained by Takata et al. (Phys. Fluids A, vol. 5, 1993, pp. 716–737) to a weakly nonlinear case. Numerical results for the drag on the basis of the Bhatnagar–Gross–Krook (BGK) model are also presented.

THE CONTRIBUTION OF THE BHATNAGAR–GROSS–KROOK MODEL TO THE DEVELOPMENT OF RAREFIED GAS DYNAMICS IN THE EARLY YEARS OF THE SPACE AGE

2013 ◽  
Vol 25 (01) ◽  
pp. 1340025
Author(s):  
RODDAM NARASIMHA
Keyword(s):  
Boltzmann Equation ◽  
Gas Dynamics ◽  
Rarefied Gas ◽  
Rarefied Gas Dynamics ◽  
Early Years ◽  
Plane Shock ◽  
Bgk Model ◽  
The Boltzmann Equation ◽  
Nonequilibrium Flows ◽  
Space Age

The advent of the space age in 1957 was accompanied by a sudden surge of interest in rarefied gas dynamics (RGD). The well-known difficulties associated with solving the Boltzmann equation that governs RGD made progress slow but the Bhatnagar–Gross–Krook (BGK) model, proposed three years before Sputnik, turned out to have been an uncannily timely, attractive and fruitful option, both for gaining insights into the Boltzmann equation and for estimating various technologically useful flow parameters. This paper gives a view of how BGK contributed to the growth of RGD during the first decade of the space age. Early efforts intended to probe the limits of the BGK model showed that, in and near both the continuum Euler limit and the collisionless Knudsen limit, BGK could provide useful answers. Attempts were therefore made to tackle more ambitious nonlinear nonequilibrium problems. The most challenging of these was the structure of a plane shock wave. The first exact numerical solutions of the BGK equation for the shock appeared during 1962 to 1964, and yielded deep insights into the character of transitional nonequilibrium flows that had resisted all attempts at solution through the Boltzmann equation. In particular, a BGK weak shock was found to be amenable to an asymptotic analysis. The results highlighted the importance of accounting separately for fast-molecule dynamics, most strikingly manifested as tails in the distribution function, both in velocity and in physical space — tails are strange versions or combinations of collisionless and collision-generated flows. However, by the mid-1960s Monte-Carlo methods of solving the full Boltzmann equation were getting to be mature and reliable and interest in the BGK waned in the following years. Interestingly, it has seen a minor revival in recent years as a tool for developing more effective algorithms in continuum computational fluid dynamics, but the insights derived from the BGK for strongly nonequilibrium flows should be of lasting value.

Effectiveness of Normal and Angled Slot Cooling

2004 ◽  
Author(s):  
A. C. Smith ◽  
J. H. Hatchett ◽  
A. C. Nix ◽  
W. F. Ng ◽  
K. A. Thole ◽  
...  
Keyword(s):  
Mach Number ◽  
Film Cooling ◽  
Fluid Dynamic ◽  
Current Experiment ◽  
Cfd Simulations ◽  
Cooling Effectiveness ◽  
Film Cooling Effectiveness ◽  
Blowing Ratio ◽  
Mach Numbers ◽  
Lift Off

An experimental and numerical investigation was conducted to determine the film cooling effectiveness of a normal slot and angled slot under realistic engine Mach number conditions. Freestream Mach numbers of 0.65 and 1.3 were tested. For the normal slot, hot gas ingestion into the slot was observed at low blowing ratios (M < 0.25). At high blowing ratios (M > 0.6) the cooling film was observed to “lift off” from the surface. For the 30° angled slot, the data was found to collapse using the blowing ratio as a scaling parameter. Results from the current experiment were compared with the subsonic data previously published. For the angle slot, at supersonic freestream Mach number, the current experiment shows that at the same x/Ms, the film-cooling effectiveness increases by as much as 25% as compared to the subsonic case. The results of the experiment also show that at the same x/Ms, the film cooling effectiveness of the angle slot is considerably higher than the normal slot, at both subsonic and supersonic Mach numbers. The flow physics for the slot tests considered here are also described with computational fluid dynamic (CFD) simulations in the subsonic and supersonic regimes.

scholarly journals A Review on BGK Models for Gas Mixtures of Mono and Polyatomic Molecules

Fluids ◽  
2021 ◽  
Vol 6 (11) ◽  
pp. 393
Author(s):  
Marlies Pirner
Keyword(s):  
Boltzmann Equation ◽  
Mathematical Theory ◽  
Degrees Of Freedom ◽  
Rarefied Gas ◽  
Entropy Inequality ◽  
Gas Mixtures ◽  
Polyatomic Molecules ◽  
Bgk Model ◽  
Practical Applications ◽  
Internal Degrees Of Freedom

We consider the Bathnagar–Gross–Krook (BGK) model, an approximation of the Boltzmann equation, describing the time evolution of a single momoatomic rarefied gas and satisfying the same two main properties (conservation properties and entropy inequality). However, in practical applications, one often has to deal with two additional physical issues. First, a gas often does not consist of only one species, but it consists of a mixture of different species. Second, the particles can store energy not only in translational degrees of freedom but also in internal degrees of freedom such as rotations or vibrations (polyatomic molecules). Therefore, here, we will present recent BGK models for gas mixtures for mono- and polyatomic particles and the existing mathematical theory for these models.

scholarly journals Simulation of Gas Flow Over a Micro Cylinder

2009 ◽  
Author(s):  
Yuan Hu ◽  
Quanhua Sun ◽  
Jing Fan
Keyword(s):  
Reynolds Number ◽  
Mach Number ◽  
Drag Coefficient ◽  
Gas Flow ◽  
Rarefied Gas ◽  
Low Reynolds Number ◽  
Pressure Drag ◽  
Low Reynolds ◽  
Rarefied Gas Effect ◽  
Gas Effect

Gas flow over a micro cylinder is simulated using both a compressible Navier-Stokes solver and a hybrid continuum/particle approach. The micro cylinder flow has low Reynolds number because of the small length scale and the low speed, which also indicates that the rarefied gas effect exists in the flow. A cylinder having a diameter of 20 microns is simulated under several flow conditions where the Reynolds number ranges from 2 to 50 and the Mach number varies from 0.1 to 0.8. It is found that the low Reynolds number flow can be compressible even when the Mach number is less than 0.3, and the drag coefficient of the cylinder increases when the Reynolds number decreases. The compressible effect will increase the pressure drag coefficient although the friction coefficient remains nearly unchanged. The rarefied gas effect will reduce both the friction and pressure drag coefficients, and the vortex in the flow may be shrunk or even disappear.

scholarly journals Hydrodynamic limits of kinetic equations for polyatomic and reactive gases

2017 ◽  
Vol 8 (1) ◽  
pp. 23-42 ◽  
Author(s):  
M. Bisi ◽  
G. Spiga
Keyword(s):  
Degrees Of Freedom ◽  
Continuum Limit ◽  
Fluid Dynamic ◽  
Energy Levels ◽  
Kinetic Equations ◽  
Navier Stokes ◽  
Bgk Model ◽  
Hydrodynamic Limits ◽  
Stress Diffusion ◽  
Reactive Gases

Abstract Starting from a kinetic BGK-model for a rarefied polyatomic gas, based on a molecular structure of discrete internal energy levels, an asymptotic Chapman-Enskog procedure is developed in the asymptotic continuum limit in order to derive consistent fluid-dynamic equations for macroscopic fields at Navier-Stokes level. In this way, the model allows to treat the gas as a mixture of mono-atomic species. Explicit expressions are given not only for dynamical pressure, but also for shear stress, diffusion velocities, and heat flux. The analysis is shown to deal properly also with a mixture of reactive gases, endowed for simplicity with translational degrees of freedom only, in which frame analogous results can be achieved.

Towards a Technique for Nonlinear Modal Analysis

2012 ◽  
Author(s):  
Simon A. Neild ◽  
Andrea Cammarano ◽  
David J. Wagg
Keyword(s):  
Normal Form ◽  
Modal Analysis ◽  
Equations Of Motion ◽  
Transformation Process ◽  
Second Order ◽  
Identity Transformation ◽  
Modal Testing ◽  
System Response ◽  
Weakly Nonlinear ◽  
Nonlinear Modal Analysis

In this paper we discuss a theoretical technique for decomposing multi-degree-of-freedom weakly nonlinear systems into a simpler form — an approach which has parallels with the well know method for linear modal analysis. The key outcome is that the system resonances, both linear and nonlinear are revealed by the transformation process. For each resonance, parameters can be obtained which characterise the backbone curves, and higher harmonic components of the response. The underlying mathematical technique is based on a near identity normal form transformation. This is an established technique for analysing weakly nonlinear vibrating systems, but in this approach we use a variation of the method for systems of equations written in second-order form. This is a much more natural approach for structural dynamics where the governing equations of motion are written in this form as standard practice. In fact the first step in the method is to carry out a linear modal transformation using linear modes as would typically done for a linear system. The near identity transform is then applied as a second step in the process and one which identifies the nonlinear resonances in the system being considered. For an example system with cubic nonlinearities, we show how the resulting transformed equations can be used to obtain a time independent representation of the system response. We will discuss how the analysis can be carried out with applied forcing, and how the approximations about response frequencies, made during the near-identity transformation, affect the accuracy of the technique. In fact we show that the second-order normal form approach can actually improve the predictions of sub- and super-harmonic responses. Finally we comment on how this theoretical technique could be used as part of a modal testing approach in future work.

Heat Transfer in Rotating Serpentine Coolant Passage With Ribbed Walls at Low Mach Numbers

2015 ◽  
Vol 7 (1) ◽  
Author(s):  
Shang-Feng Yang ◽  
Je-Chin Han ◽  
Salam Azad ◽  
Ching-Pang Lee
Keyword(s):  
Heat Transfer ◽  
Mach Number ◽  
Reynolds Numbers ◽  
Working Fluid ◽  
Transfer Coefficients ◽  
Low Mach Number ◽  
Heat Transfer Coefficients ◽  
Rotation Numbers ◽  
Wide Range ◽  
Mach Numbers

This paper experimentally investigates the effect of rotation on heat transfer in typical turbine blade serpentine coolant passage with ribbed walls at low Mach numbers. To achieve the low Mach number (around 0.01) condition, pressurized Freon R-134a vapor is utilized as the working fluid. The flow in the first passage is radial outward, after the 180 deg tip turn the flow is radial inward to the second passage, and after the 180 deg hub turn the flow is radial outward to the third passage. The effects of rotation on the heat transfer coefficients were investigated at rotation numbers up to 0.6 and Reynolds numbers from 30,000 to 70,000. Heat transfer coefficients were measured using the thermocouples-copper-plate-heater regional average method. Heat transfer results are obtained over a wide range of Reynolds numbers and rotation numbers. An increase in heat transfer rates due to rotation is observed in radially outward passes; a reduction in heat transfer rate is observed in the radially inward pass. Regional heat transfer coefficients are correlated with Reynolds numbers for nonrotation and with rotation numbers for rotating condition, respectively. The results can be useful for understanding real rotor blade coolant passage heat transfer under low Mach number, medium–high Reynolds number, and high rotation number conditions.

Effect of Mach number on the acoustic field of 2:1 elliptic-slot jet

2001 ◽  
Vol 105 (1043) ◽  
pp. 9-16 ◽  
Author(s):  
S. B. Verma ◽  
E. Rathakrishnan
Keyword(s):  
Mach Number ◽  
Fundamental Frequency ◽  
Acoustic Field ◽  
Major Axis ◽  
Shock Structure ◽  
Minor Axis ◽  
Noise Levels ◽  
Underexpanded Jets ◽  
Barrel Shock ◽  
Mach Numbers

Abstract The shock-structure and the related acoustic field of underexpanded jets undergoes significant changes as the Mach number Mj is increased. The present investigation is carried out to study the effect of Mach number on an underexpanded 2:1 elliptic-slot jet. Experimental data are presented for fully expanded Mach numbers ranging from 1.3 to 2.0. It is observed that the ‘cross-over’ point at the end of the first cell at low Mach numbers gets replaced by a normal shock at a highly underexpanded condition resulting in the formation of a ‘barrel’ shock along the minor-axis side with a ‘bulb’ shock formed along the major-axis side. The above change in shock structure is accompanied by a related change in the acoustic field. The amplitude of fundamental frequency along the minor-axis side grows with Mj but falls beyond Mj = 1.75. Along the major-axis side, however, the fundamental frequency does not exist at low Mach numbers. It appears at Mj = 1.75 but then falls at Mj = 2.0. The related azimuthal directivity of overall noise levels (OASPL) shows significant changes with Mj.

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