scholarly journals Unsteady turbulent line plumes

2018 ◽  
Vol 856 ◽  
pp. 103-134
Author(s):  
Andrew J. Hogg ◽  
Edward J. Goldsmith ◽  
Mark J. Woodhouse
Keyword(s):  
Numerical Solutions ◽  
Similarity Solutions ◽  
Stable Model ◽  
Governing Equations ◽  
Shape Factors ◽  
Piecewise Constant ◽  
Horizontal Variation ◽  
Model Account ◽  
Ill Posed ◽  
Well Posed

The unsteady ascent of a buoyant, turbulent line plume through a quiescent, uniform environment is modelled in terms of the width-averaged vertical velocity and density deficit. It is demonstrated that for a well-posed, linearly stable model, account must be made for the horizontal variation of the velocity and the density deficit; in particular the variance of the velocity field and the covariance of the density deficit and velocity fields, represented through shape factors, must exceed threshold values, and that models based upon ‘top-hat’ distributions in which the dependent fields are piecewise constant are ill-posed. Numerical solutions of the nonlinear governing equations are computed to reveal that the transient response of the system to an instantaneous change in buoyancy flux at the source may be captured through new similarity solutions, the form of which depend upon both the ratio of the old to new buoyancy fluxes and the shape factors.

Heat Transfer in a Laminar Channel Flow Generated by Injection Through Porous Walls

2007 ◽  
Vol 129 (8) ◽  
pp. 1048-1057 ◽  
Author(s):  
Clarisse Fournier ◽  
Marc Michard ◽  
Françoise Bataille
Keyword(s):  
Channel Flow ◽  
Numerical Solutions ◽  
Scaling Law ◽  
Similarity Solutions ◽  
Temperature Profiles ◽  
Governing Equations ◽  
Temperature Boundary ◽  
Laminar Channel Flow ◽  
Porous Walls ◽  
Uniform Fluid

Steady state similarity solutions are computed to determine the temperature profiles in a laminar channel flow driven by uniform fluid injection at one or two porous walls. The temperature boundary conditions are non-symmetric. The numerical solution of the governing equations permit to analyze the influence of the governing parameters, the Reynolds and Péclet numbers. For both geometries, we deduce a scaling law for the boundary layer thickness as a function of the Péclet number. We also compare the numerical solutions with asymptotic expansions in the limit of large Péclet numbers. Finally, for non-symmetric injection, we derive from the computed temperature profile a relationship between the Nusselt and Péclet numbers.

Laminar Free Convection on a Curved Wall

1971 ◽  
Vol 38 (4) ◽  
pp. 1081-1083
Author(s):  
K. W. McAlister
Keyword(s):  
Heat Transfer ◽  
Free Convection ◽  
Temperature Variation ◽  
Newtonian Fluid ◽  
Numerical Solutions ◽  
Similarity Solutions ◽  
Governing Equations ◽  
Transfer Rates ◽  
Arbitrary Temperature ◽  
Free Parameters

Laminar free convection of a Newtonian fluid passing over a curved wall having arbitrary temperature variation is considered. The governing equations are presented and the method of free parameters is used to investigate the existence of similarity solutions. It is found that similarity solutions do exist when the wall inclination and temperature are required to be certain functions of the coordinate parallel to the wall. Numerical solutions to several example cases are presented which indicate that higher heat-transfer rates are possible on a wall which is concave with respect to the fluid.

scholarly journals Partial regularisation of the incompressible 𝜇(I)-rheology for granular flow

2017 ◽  
Vol 828 ◽  
pp. 5-32 ◽  
Author(s):  
T. Barker ◽  
J. M. N. T. Gray
Keyword(s):  
High Speed ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Functional Dependence ◽  
Granular Flows ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Major Step ◽  
Ill Posed ◽  
Well Posed

In recent years considerable progress has been made in the continuum modelling of granular flows, in particular the $\unicode[STIX]{x1D707}(I)$-rheology, which links the local viscosity in a flow to the strain rate and pressure through the non-dimensional inertial number $I$. This formulation greatly benefits from its similarity to the incompressible Navier–Stokes equations as it allows many existing numerical methods to be used. Unfortunately, this system of equations is ill posed when the inertial number is too high or too low. The consequence of ill posedness is that the growth rate of small perturbations tends to infinity in the high wavenumber limit. Due to this, numerical solutions are grid dependent and cannot be taken as being physically realistic. In this paper changes to the functional form of the $\unicode[STIX]{x1D707}(I)$ curve are considered, in order to maximise the range of well-posed inertial numbers, while preserving the overall structure of the equations. It is found that when the inertial number is low there exist curves for which the equations are guaranteed to be well posed. However when the inertial number is very large the equations are found to be ill posed regardless of the functional dependence of $\unicode[STIX]{x1D707}$ on $I$. A new $\unicode[STIX]{x1D707}(I)$ curve, which is inspired by the analysis of the governing equations and by experimental data, is proposed here. In order to test this regularised rheology, transient granular flows on inclined planes are studied. It is found that simulations of flows, which show signs of ill posedness with unregularised models, are numerically stable and match key experimental observations when the regularised model is used. This paper details two-dimensional transient computations of decelerating flows where the inertial number tends to zero, high-speed flows that have large inertial numbers, and flows which develop into granular rollwaves. This is the first time that granular rollwaves have been simulated in two dimensions, which represents a major step towards the simulation of other complex granular flows.

Non-similarity solutions to the corner boundary-layer equations (and the effects of wall transpiration)

1999 ◽  
Vol 400 ◽  
pp. 125-162 ◽  
Author(s):  
PETER W. DUCK ◽  
SIMON R. STOW ◽  
MANHAR R. DHANAK
Keyword(s):  
Boundary Layer ◽  
Numerical Solutions ◽  
A Priori ◽  
Leading Edge ◽  
Flow Region ◽  
Similarity Solutions ◽  
Leading Order ◽  
Governing Equations ◽  
Boundary Layer Equations ◽  
Pressure Term

The incompressible boundary layer in the corner formed by two intersecting, semi-infinite planes is investigated, when the free-stream flow, aligned with the corner, is taken to be of the form U∞F(x), x representing the non-dimensional streamwise distance from the leading edge. In Dhanak & Duck (1997) similarity solutions for F(x) = xn were considered, and it was found that solutions exist for only a range of values of n, whilst for ∞ > n > −0.018, approximately, two solutions exist. In this paper, we extend the work of Dhanak & Duck to the case of non-90° corner angles and allow for streamwise development of solutions. In addition, the effect of transpiration at the walls of the corner is investigated. The governing equations are of boundary-layer type and as such are parabolic in nature. Crucially, although the leading-order pressure term is known a priori, the third-order pressure term is not, but this is nonetheless present in the leading-order governing equations, together with the transverse and crossflow viscous terms.Particular attention is paid to flows which develop spatially from similarity solutions. It turns out that two scenarios are possible. In some cases the problem may be treated in the usual parabolic sense, with standard numerical marching procedures being entirely appropriate. In other cases standard marching procedures lead to numerically inconsistent solutions. The source of this difficulty is linked to the existence of eigensolutions emanating from the leading edge (which are not present in flows appropriate to the first scenario), analogous to those found in the computation of some two-dimensional hypersonic boundary layers (Neiland 1970; Mikhailov et al. 1971; Brown & Stewartson 1975). In order to circumvent this difficulty, a different numerical solution strategy is adopted, based on a global Newton iteration procedure.A number of numerical solutions for the entire corner flow region are presented.

On the Interaction and Reflection of Shocks in Hyperelastic Strings

1991 ◽  
Vol 58 (2) ◽  
pp. 554-558 ◽  
Author(s):  
J. L. Wegner ◽  
L. Jiang ◽  
J. B. Haddow
Keyword(s):  
Method Of Characteristics ◽  
Numerical Solutions ◽  
Similarity Solutions ◽  
Finite Amplitude ◽  
Governing Equations ◽  
Wave Interactions ◽  
Amplitude Wave ◽  
Multiple Wave ◽  
The Method Of Characteristics ◽  
Interaction Problem

Governing equations for finite amplitude wave propagation in stretched hyperelastic strings are given in recent papers, (Beatty and Haddow, 1985), along with similarity solutions for symmetrically plucked and impacted strings. The similarity solutions are valid until the first reflections at the fixed ends and in this paper we consider symmetrically plucked Mooney-Rivlin strings and investigate the response after reflections. The method of characteristics is applied to extend the results of the similarity solutions and to obtain solutions for the interaction of a reflected longitudinal shock and incident transverse shock and the reflection of an incident transverse shock. A deformed shape, which is not intuitively obvious, is predicted by the solution of the interaction problem and is confirmed by an experimental study. A finite difference scheme is used to obtain numerical solutions, which are valid after multiple wave interactions and reflections occur. Solutions obtained by the method of characteristics are used as a partial check on the numerical results.

scholarly journals Unsteady turbulent buoyant plumes

2016 ◽  
Vol 794 ◽  
pp. 595-638 ◽  
Author(s):  
M. J. Woodhouse ◽  
J. C. Phillips ◽  
A. J. Hogg
Keyword(s):  
Similarity Solution ◽  
Axial Velocity ◽  
Shape Factor ◽  
Numerical Solutions ◽  
Integral Model ◽  
Governing Equations ◽  
Shape Factors ◽  
Buoyant Plumes ◽  
Radial Profiles ◽  
Steady Solutions

We model the unsteady evolution of turbulent buoyant plumes following temporal changes to the source conditions. The integral model is derived from radial integration of the governing equations expressing the evolution of mass, axial momentum and buoyancy in the plume. The non-uniform radial profiles of the axial velocity and density deficit in the plume are explicitly captured by shape factors in the integral equations; the commonly assumed top-hat profiles lead to shape factors equal to unity. The resultant model for unsteady plumes is hyperbolic when the momentum shape factor, determined from the radial profile of the mean axial velocity in the plume, differs from unity. The solutions of the model when source conditions are maintained at constant values are shown to retain the form of the well-established steady plume solutions. We demonstrate through a linear stability analysis of these steady solutions that the inclusion of a momentum shape factor in the governing equations that differs from unity leads to a well-posed integral model. Therefore, our model does not exhibit the mathematical pathologies that appear in previously proposed unsteady integral models of turbulent plumes. A stability threshold for the value of the shape factor is also identified, resulting in a range of its values where the amplitudes of small perturbations to the steady solutions decay with distance from the source. The hyperbolic character of the system of equations allows the formation of discontinuities in the fields describing the plume properties during the unsteady evolution, and we compute numerical solutions to illustrate the transient development of a plume following an abrupt change in the source conditions. The adjustment of the plume to the new source conditions occurs through the propagation of a pulse of fluid through the plume. The dynamics of this pulse is described by a similarity solution and, through the construction of this new similarity solution, we identify three regimes in which the evolution of the transient pulse following adjustment of the source qualitatively differs.

Convergence and stability analysis of the half thresholding based few-view CT reconstruction

2020 ◽  
Vol 28 (6) ◽  
pp. 829-847
Author(s):  
Hua Huang ◽  
Chengwu Lu ◽  
Lingli Zhang ◽  
Weiwei Wang
Keyword(s):  
Noise Suppression ◽  
Reconstructed Image ◽  
Reference Image ◽  
Projection Data ◽  
Ct Reconstruction ◽  
Reconstruction Algorithms ◽  
Ill Posed ◽  
Thresholding Algorithm ◽  
The Stability ◽  
Well Posed

AbstractThe projection data obtained using the computed tomography (CT) technique are often incomplete and inconsistent owing to the radiation exposure and practical environment of the CT process, which may lead to a few-view reconstruction problem. Reconstructing an object from few projection views is often an ill-posed inverse problem. To solve such problems, regularization is an effective technique, in which the ill-posed problem is approximated considering a family of neighboring well-posed problems. In this study, we considered the {\ell_{1/2}} regularization to solve such ill-posed problems. Subsequently, the half thresholding algorithm was employed to solve the {\ell_{1/2}} regularization-based problem. The convergence analysis of the proposed method was performed, and the error bound between the reference image and reconstructed image was clarified. Finally, the stability of the proposed method was analyzed. The result of numerical experiments demonstrated that the proposed method can outperform the classical reconstruction algorithms in terms of noise suppression and preserving the details of the reconstructed image.

scholarly journals Elastostatics of Bernoulli–Euler Beams Resting on Displacement-Driven Nonlocal Foundation

Nanomaterials ◽  
2021 ◽  
Vol 11 (3) ◽  
pp. 573
Author(s):  
Marzia Sara Vaccaro ◽  
Francesco Paolo Pinnola ◽  
Francesco Marotti de Sciarra ◽  
Raffaele Barretta
Keyword(s):  
Boundary Conditions ◽  
Structural Engineering ◽  
Beam Deflection ◽  
Soil Reaction ◽  
Differential Problem ◽  
Reaction Field ◽  
Ill Posed ◽  
Elastic Responses ◽  
Benchmark Solutions ◽  
Well Posed

The simplest elasticity model of the foundation underlying a slender beam under flexure was conceived by Winkler, requiring local proportionality between soil reactions and beam deflection. Such an approach leads to well-posed elastostatic and elastodynamic problems, but as highlighted by Wieghardt, it provides elastic responses that are not technically significant for a wide variety of engineering applications. Thus, Winkler’s model was replaced by Wieghardt himself by assuming that the beam deflection is the convolution integral between soil reaction field and an averaging kernel. Due to conflict between constitutive and kinematic compatibility requirements, the corresponding elastic problem of an inflected beam resting on a Wieghardt foundation is ill-posed. Modifications of the original Wieghardt model were proposed by introducing fictitious boundary concentrated forces of constitutive type, which are physically questionable, being significantly influenced on prescribed kinematic boundary conditions. Inherent difficulties and issues are overcome in the present research using a displacement-driven nonlocal integral strategy obtained by swapping the input and output fields involved in Wieghardt’s original formulation. That is, nonlocal soil reaction fields are the output of integral convolutions of beam deflection fields with an averaging kernel. Equipping the displacement-driven nonlocal integral law with the bi-exponential averaging kernel, an equivalent nonlocal differential problem, supplemented with non-standard constitutive boundary conditions involving nonlocal soil reactions, is established. As a key implication, the integrodifferential equations governing the elastostatic problem of an inflected elastic slender beam resting on a displacement-driven nonlocal integral foundation are replaced with much simpler differential equations supplemented with kinematic, static, and new constitutive boundary conditions. The proposed nonlocal approach is illustrated by examining and analytically solving exemplar problems of structural engineering. Benchmark solutions for numerical analyses are also detected.

Thermal Post-Buckling Analysis of Functionally Graded Composite Plates with Nonlinear Aerodynamic Forces

2010 ◽  
Vol 123-125 ◽  
pp. 280-283
Author(s):  
Chang Yull Lee ◽  
Ji Hwan Kim
Keyword(s):  
Material Properties ◽  
Numerical Solutions ◽  
Virtual Work ◽  
Shear Deformation Theory ◽  
Composite Plates ◽  
Buckling Analysis ◽  
Functionally Graded ◽  
Governing Equations ◽  
Functionally Graded Composite ◽  
Post Buckling

The post-buckling of the functionally graded composite plate under thermal environment with aerodynamic loading is studied. The structural model has three layers with ceramic, FGM and metal, respectively. The outer layers of the sandwich plate are different homogeneous and isotropic material properties for ceramic and metal. Whereas the core is FGM layer, material properties vary continuously from one interface to the other in the thickness direction according to a simple power law distribution in terms of the volume fractions. Governing equations are derived by using the principle of virtual work and numerical solutions are solved through a finite element method. The first-order shear deformation theory and von-Karman strain-displacement relations are based to derive governing equations of the plate. Aerodynamic effects are dealt by adopting nonlinear third-order piston theory for structural and aerodynamic nonlinearity. The Newton-Raphson iterative method applied for solving the nonlinear equations of the thermal post-buckling analysis

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