CONVERGENCE OF LOCALLY SELFSIMILAR SOLUTIONS TO EXACT NUMERICAL SOLUTIONS OF BOUNDARY LAYER EQUATIONS FOR A PLATE

2021 ◽  
pp. 49-62
Author(s):  
Yu.N. Grigoriev ◽  
◽  
A.G. Gorobchuk ◽  
I.V. Ershov ◽  
◽  
...  
Keyword(s):  
Boundary Layer ◽  
Finite Difference ◽  
Linear Stability ◽  
Numerical Solutions ◽  
Linear Stability Theory ◽  
Plane Boundary ◽  
Transfer Coefficients ◽  
Vibrationally Excited ◽  
Self Similar ◽  
Similar Solutions

This paper considers a possibility of using locally self-similar solutions for a stationary boundary layer in linear stability problems. The solutions, obtained at various boundary conditions for a vibrationally excited gas, are compared with finite-difference calculations of the corresponding flows. An initial system of equations for a plane boundary layer of the vibrationally excited gas is derived from complete equations of two-temperature relaxation aerodynamics. Relaxation of vibrational modes of gas molecules is described in the framework of the Landau – Teller equation. Transfer coefficients depend on the static flow temperature. In a complete problem statement, the flows are calculated using the Crank – Nicolson finite-difference scheme. In all the considered cases, it is shown that the locally self-similar velocity and temperature profiles converge to the corresponding profiles for a fully developed boundary-layer flow calculated in a finite-difference formulation. The obtained results justify the use of locally self-similar solutions in problems of the linear stability theory for boundary-layer flows of a vibrationally excited gas.

Расчеты сверхзвукового пограничного слоя в полной и локально автомодельной постановках

2020 ◽  
Author(s):  
Y.N. Grigoryev ◽  
A.G. Gorobchuk ◽  
I.V. Ershov
Keyword(s):  
Boundary Layer ◽  
Finite Difference ◽  
Initial Conditions ◽  
Plane Boundary ◽  
Transfer Coefficients ◽  
Vibrationally Excited ◽  
Limit Values ◽  
Various Boundary Conditions ◽  
Self Similar ◽  
Similar Solutions

The article considers the possibility of using locally self-similar solutions of a stationary boundary layer in linear stability problems. These solutions are compared with various boundary conditions for a vibrationally excited gas with finite-difference calculations of the corresponding flows. The initial system of equations for a plane boundary layer of a vibrationally excited gas was obtained from the complete equations of two-temperature relaxation aerodynamics. The relaxation of vibrational modes of gas molecules is described in the framework of the LandauTeller equation. Transfer coefficients depend on the static flow temperature. It is shown that in all considered cases the convergence of profiles of hydrodynamic variables to some limit values takes place for the longitudinal coordinate 8 . . . 15. In parallel, the same flows were calculated using the full formulation based on the finite-difference KrankNicholson type scheme. It is shown that for all considered boundary and initial conditions the limiting locally self-similar profiles coincide with the profiles calculated within the full formulation. The obtained result substantiates the use of locally self-similar solutions in problems of the linear theory of stability of boundary layer flows of vibrationally excited gas. Проведены расчеты течения в плоском пограничном слое сжимаемого колебательно возбужденного газа в локально автомодельной постановке для ряда характерных условий внешнего потока и теплообмена на границе. Показано, что во всех рассмотренных случаях имеет место сходимость профилей гидродинамических переменных к некоторым предельным значениям для продольной координаты x> 8 . . . 15. Параллельно те же течения рассчитывались в полной постановке на основе конечно-разностной схемы типа Кранка-Николсон. Показано, что для всех рассмотренных граничных и начальных условий предельные локально автомодельные профили совпадают с профилями, рассчитанными в полной постановке. Это позволяет обоснованно использовать легко рассчитываемые локально автомодельные профили в задачах линейной теории устойчивости.

Self-similar solutions to the second-order incompressible boundary-layer equations

1970 ◽  
Vol 40 (2) ◽  
pp. 343-360 ◽  
Author(s):  
M. J. Werle ◽  
R. T. Davis
Keyword(s):  
Boundary Layer ◽  
Numerical Solutions ◽  
Second Order ◽  
Closed Form Solutions ◽  
Boundary Layer Equations ◽  
Incompressible Boundary Layer ◽  
Incompressible Boundary ◽  
Self Similar ◽  
Displacement Speed ◽  
Similar Solutions

Solutions are obtained for the self-similar form of the incompressible boundary-layer equations for all four second-order contributors, i.e. vorticity interaction, displacement speed, longitudinal and transverse curvature. These results are found to contain all previous self-similar solutions as members of the much larger family of solutions presented here. Numerical solutions are presented for a large number of cases, and several closed form solutions, which may have special significance for the separation problem, are also discussed.

Stability of vertical natural convection boundary layers: some numerical solutions

1971 ◽  
Vol 48 (4) ◽  
pp. 625-646 ◽  
Author(s):  
C. A. Hieber ◽  
B. Gebhart
Keyword(s):  
Boundary Layer ◽  
Natural Convection ◽  
Linear Stability ◽  
Stability Theory ◽  
Numerical Solutions ◽  
Numerical Procedure ◽  
Thermal Capacity ◽  
Linear Stability Theory ◽  
Neutral Stability ◽  
Convection Boundary

Linear stability theory is applied to the natural convection boundary layer arising from a vertical plate dissipating a uniform heat flux. By using a numerical procedure which is much simpler than those previously employed on this problem, computer solutions are obtained for a much larger range of the Grashof number (G). For a Prandtl number (σ) of 0·733, it is found that, asG→ ∞: the effect of temperature coupling vanishes more rapidly than that of viscosity; the upper branch of the neutral curve is oscillatory but does approach a finite non-zero inviscid asymptote. For moderate and large values of σ, a loop appears in the neutral stability curve as a result of the merging of two unstable modes. As σ → ∞, the mode associated with the uncoupled (i.e. Orr–Sommerfeld) problem rapidly becomes less unstable than that arising from the temperature coupling, with the stability characteristics being independent of the thermal capacity of the plate. For small values of σ, only one unstable mode is found to exist with the coupling effect being negligible in the case of large thermal capacity plates but markedly destabilizing when the thermal capacity is small.By obtaining numerical results out toG≈ 1010for the cases σ = 0·733 and 6·7, it becomes possible to attempt to directly relate the theory to the actual observance of turbulent transition. Based upon comparison with available experimental data, empirical correlations are obtained between the linear stability theory and the régimes in which: (i) the boundary layer is first noticeably oscillatory; (ii) the mean (temporal) flow quantities first deviate significantly from those of laminar flow.

Comparison of the numerical and self-similar solutions of Sedov’s problem on a point explosion in gas

2020 ◽  
Vol 15 (3-4) ◽  
pp. 212-216
Author(s):  
R.Kh. Bolotnova ◽  
V.A. Korobchinskaya
Keyword(s):  
Gas Dynamics ◽  
Numerical Solutions ◽  
Similarity Theory ◽  
Compressible Medium ◽  
Initial Moment ◽  
Point Explosion ◽  
Perfect Gas ◽  
Gas Dynamics Equations ◽  
Self Similar ◽  
Similar Solutions

Comparative analysis of solutions of Sedov’s problem of a point explosion in gas for the plane case, obtained by the analytical method and using the open software package of computational fluid dynamics OpenFOAM, is carried out. A brief analysis of methods of dimensionality and similarity theory used for the analytical self-similar solution of point explosion problem in a perfect gas (nitrogen) which determined by the density of uncompressed gas, magnitude of released energy, ratio of specific heat capacities and by the index of geometry of the explosion is given. The system of one-dimensional gas dynamics equations for a perfect gas includes the laws of conservation of mass, momentum, and energy is used. It is assumed that at the initial moment of time there is a point explosion with instantaneous release of energy. Analytical self-similar solutions for the Euler and Lagrangian coordinates, mass velocity, pressure, temperature, and density in the case of plane geometry are given. The numerical simulation of considered process in sonicFoam solver of OpenFOAM package built on the PISO algorithm was performed. For numerical modeling the system of differential equations of gas dynamics is used, including the equations of continuity, Navier-Stokes motion for a compressible medium and conservation of internal energy. Initial and boundary conditions were selected in accordance with the obtained analytical solution using the setFieldsDict, blockMeshDict, and uniformFixedValue utilities. The obtained analytical and numerical solutions have a satisfactory agreement.

Natural convection flow far from a horizontal plate

1999 ◽  
Vol 387 ◽  
pp. 227-254 ◽  
Author(s):  
VALOD NOSHADI ◽  
WILHELM SCHNEIDER
Keyword(s):  
Boundary Layer ◽  
Prandtl Number ◽  
Stokes Equations ◽  
Axisymmetric Flow ◽  
The Self ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Boundary Layer Equations ◽  
Self Similar ◽  
Similar Solutions

Plane and axisymmetric (radial), horizontal laminar jet flows, produced by natural convection on a horizontal finite plate acting as a heat dipole, are considered at large distances from the plate. It is shown that physically acceptable self-similar solutions of the boundary-layer equations, which include buoyancy effects, exist in certain Prandtl-number regimes, i.e. 0.5<Pr[les ]1.470588 for plane, and Pr>1 for axisymmetric flow. In the plane flow case, the eigenvalues of the self-similar solutions are independent of the Prandtl number and can be determined from a momentum balance, whereas in the axisymmetric case the eigenvalues depend on the Prandtl number and are to be determined as part of the solution of the eigenvalue problem. For Prandtl numbers equal to, or smaller than, the lower limiting values of 0.5 and 1 for plane and axisymmetric flow, respectively, the far flow field is a non-buoyant jet, for which self-similar solutions of the boundary-layer equations are also provided. Furthermore it is shown that self-similar solutions of the full Navier–Stokes equations for axisymmetric flow, with the velocity varying as 1/r, exist for arbitrary values of the Prandtl number.Comparisons with finite-element solutions of the full Navier–Stokes equations show that the self-similar boundary-layer solutions are asymptotically approached as the plate Grashof number tends to infinity, whereas the self-similar solution to the full Navier–Stokes equations is applicable, for a given value of the Prandtl number, only to one particular, finite value of the Grashof number.In the Appendices second-order boundary-layer solutions are given, and uniformly valid composite expansions are constructed; asymptotic expansions for large values of the lateral coordinate are performed to study the decay of the self-similar boundary-layer flows; and the stability of the jets is investigated using transient numerical solutions of the Navier–Stokes equations.

Linear Stability of Momentum Boundary Layer Flow and Heat Transfer Over a Moving Wedge

2020 ◽  
Vol 142 (6) ◽  
Author(s):  
Ramesh B. Kudenatti ◽  
Noor E. Misbah ◽  
M. C. Bharathi
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Linear Stability ◽  
Boundary Layer Flow ◽  
Shooting Methods ◽  
Flow And Heat Transfer ◽  
Moving Wedge ◽  
Layer Flow ◽  
Steady Solutions ◽  
Similar Solutions

Abstract This paper studies the linear stability of the unsteady boundary-layer flow and heat transfer over a moving wedge. Both mainstream flow outside the boundary layer and the wedge velocities are approximated by the power of the distance along the wedge wall. In a similar manner, the temperature of the wedge is approximated by the power of the distance that leads to a wall exponent temperature parameter. The governing boundary layer equations admit a class of self-similar solutions under these approximations. The Chebyshev collocation and shooting methods are utilized to predict the upper and lower branch solutions for various parameters. For these two solutions, the velocity, temperature profiles, wall shear-stress, and temperature gradient are entirely different and need to be assessed for their stability as to which of these solutions is practically realizable. It is shown that algebraically growing steady solutions do exist and their effects are significant in the unsteady context. The resulting eigenvalue problem determines whether or not the steady solutions are stable. There are interesting results that are linked to bypass an important class of boundary layer flow and heat transfer. The hydrodynamics behind these results are discussed in some detail.

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