Thermomechanical Equations Governing a Material With Prescribed Temperature-Dependent Density With Application to Nonisothermal Plane Poiseuille Flow

1996 ◽  
Vol 63 (4) ◽  
pp. 1011-1018 ◽  
Author(s):  
D. Cao ◽  
S. E. Bechtel ◽  
M. G. Forest
Keyword(s):  
Poiseuille Flow ◽  
Ad Hoc ◽  
Three Dimensional ◽  
Incompressible Material ◽  
Momentum Equation ◽  
Plane Poiseuille Flow ◽  
Temperature Dependent ◽  
Incompressibility Constraint ◽  
Expansion Cooling ◽  
Dependent Density

The standard practice in the literature for modeling materials processing in which changes in temperature induce significant volume changes is based on the a posteriori substitution of a temperature-dependent expression for density into the governing equations for an incompressible material. In this paper we show this ad hoc approach misses important terms in the equations, and by example show the ad hoc equations fail to capture important physical effects. First we derive the three-dimensional equations which govern the deformation and heat transfer of materials with prescribed temperature-dependent density. Specification of density as a function of temperature translates to a thermomechanical constraint, in contrast to the purely mechanical incompressibility constraint, so that the constraint response function (“pressure”) enters into the energy equation as well as the momentum equation. Then we demonstrate the effect of the correct constraint response by comparing solutions of our thermomechanical theory with solutions of the ad hoc theory in plane Poiseuille flow. The differences are significant, both quantitatively and qualitatively. In particular, the observed phenomenon of expansion cooling is captured by the thermomechanically constrained theory, but not by the ad hoc theory.

The initial-value problem for three-dimensional disturbances in plane Poiseuille flow of helium II

2008 ◽  
Vol 598 ◽  
pp. 227-244 ◽  
Author(s):  
LARS B. BERGSTRÖM
Keyword(s):  
Poiseuille Flow ◽  
Three Dimensional ◽  
Peak Level ◽  
Minor Effect ◽  
Transient Growth ◽  
Plane Poiseuille Flow ◽  
Mean Flow ◽  
Normal Fluid ◽  
Helium Ii ◽  
Fluid Field

The time development of small three-dimensional disturbances in plane Poiseuille flow of helium II is considered. The study is conducted by considering the interaction of a normal fluid field and a superfluid field. The interaction is caused by a mutual friction forcing between the two flow fields. Specifically, the stability of the normal fluid affected by the mutual forcing is considered. Compared to the ordinary fluid case where the mutual forcing is not present, the presence of the mutual forcing implies a substantial increase of the transient growth of the disturbances. The increase of the transient growth occurs because the mutual forcing reduces the damping of the disturbances. The phase of transient growth becomes thereby more prolonged and higher levels of amplification are reached. There is also a minor effect on the transient growth caused by the modification of the mean flow owing to the mutual forcing. The strongest transient growth occurs for streamwise elongated disturbances, i.e. disturbances with streamwise wavenumber α = 0. When α increases beyond zero, the transient amplification quickly becomes reduced. Striking differences compared to the ordinary fluid case are that the largest transient amplification does not occur when the spanwise wavenumber (β) is close to two and that the peak level of the disturbance energy density amplification does not depend on the square of the Reynolds number.

Energy growth of three-dimensional disturbances in plane Poiseuille flow

1991 ◽  
Vol 224 ◽  
pp. 241-260 ◽  
Author(s):  
L. Hårkan Gustavsson
Keyword(s):  
Energy Density ◽  
Poiseuille Flow ◽  
Initial Conditions ◽  
Formal Solution ◽  
Three Dimensional ◽  
Propagation Speed ◽  
Inhomogeneous Equation ◽  
Normal Velocity ◽  
Plane Poiseuille Flow ◽  
Initial Value

The development of a small three-dimensional disturbance in plane Poiseuille flow is considered. Its kinetic energy is expressed in terms of the velocity and vorticity components normal to the wall. The normal vorticity develops according to the mechanism of vortex stretching and is described by an inhomogeneous equation, where the spanwise variation of the normal velocity acts as forcing. To study specifically the effect of the forcing, the initial normal vorticity is set to zero and the energy density in the wavenumber plane, induced by the normal velocity, is determined. In particular, the response from individual (and damped) Orr–Sommerfeld modes is calculated, on the basis of a formal solution to the initial-value problem. The relevant timescale for the development of the perturbation is identified as a viscous one. Even so, the induced energy density can greatly exceed that associated with the initial normal velocity, before decay sets in. Initial conditions corresponding to the least-damped Orr–Sommerfeld mode induce the largest energy density and a maximum is obtained for structures infinitely elongated in the streamwise direction. In this limit, the asymptotic solution is derived and it shows that the spanwise wavenumbers at which the largest amplification occurs are 2.60 and 1.98, for symmetric and antisymmetric normal vorticity, respectively. The asymptotic analysis also shows that the propagation speed for induced symmetric vorticity is confined to a narrower range than that for antisymmetric vorticity. From a consideration of the neglected nonlinear terms it is found that the normal velocity component cannot be nonlinearly affected by the normal vorticity growth for structures with no streamwise dependence.

scholarly journals The stability of thermally stratified plane Poiseuille flow

1968 ◽  
Vol 33 (1) ◽  
pp. 21-32 ◽  
Author(s):  
K. S. Gage ◽  
W. H. Reid
Keyword(s):  
Poiseuille Flow ◽  
Richardson Number ◽  
Three Dimensional ◽  
Complete Analysis ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Plane Poiseuille Flow ◽  
Spiral Flow ◽  
Tollmien Schlichting ◽  
The Stability

In studying the stability of a thermally stratified fluid in the presence of a viscous shear flow, we have a situation in which there is an important interaction between the mechanism of instability due to the stratification and the Tollmien-Schlichting mechanism due to the shear. A complete analysis has been carried out for the Bénard problem in the presence of a plane Poiseuille flow and it is shown that, although Squire's transformation can be used to reduce the three-dimensional problem to an equivalent two-dimensional one, a theorem of Squire's type does not follow unless the Richardson number exceeds a certain small negative value. This conclusion follows from the fact that, when the stratification is unstable and the Prandtl number is unity, the equivalent two-dimensional problem becomes identical mathematically to the stability problem for spiral flow between rotating cylinders and, from the known results for the spiral flow problem, Squire's transformation can then be used to obtain the complete three-dimensional stability boundary. For the case of stable stratification, however, Squire's theorem is valid and the instability is of the usual Tollmien—Schlichting type. Additional calculations have been made for this case which show that the flow is completely stabilized when the Richardson number exceeds a certain positive value.

scholarly journals Localized vortex/Tollmien–Schlichting wave interaction states in plane Poiseuille flow

2016 ◽  
Vol 791 ◽  
pp. 97-121 ◽  
Author(s):  
L. J. Dempsey ◽  
K. Deguchi ◽  
P. Hall ◽  
A. G. Walton
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Wave Interaction ◽  
Travelling Wave ◽  
Navier Stokes ◽  
Spanwise Direction ◽  
Plane Poiseuille Flow ◽  
Tollmien Schlichting

Strongly nonlinear three-dimensional interactions between a roll–streak structure and a Tollmien–Schlichting wave in plane Poiseuille flow are considered in this study. Equations governing the interaction at high Reynolds number originally derived by Bennett et al. (J. Fluid Mech., vol. 223, 1991, pp. 475–495) are solved numerically. Travelling wave states bifurcating from the lower branch linear neutral point are tracked to finite amplitudes, where they are observed to localize in the spanwise direction. The nature of the localization is analysed in detail near the relevant spanwise locations, revealing the presence of a singularity which slowly develops in the governing interaction equations as the amplitude of the motion is increased. Comparisons with the full Navier–Stokes equations demonstrate that the finite Reynolds number solutions gradually approach the numerical asymptotic solutions with increasing Reynolds number.

Flow structures in spanwise rotating plane Poiseuille flow based on thermal analogy

2021 ◽  
Vol 933 ◽  
Author(s):  
Shengqi Zhang ◽  
Zhenhua Xia ◽  
Shiyi Chen
Keyword(s):  
Poiseuille Flow ◽  
Large Scale ◽  
Reynolds Shear Stress ◽  
Three Dimensional ◽  
Streamwise Velocity ◽  
Flow Structures ◽  
Plane Poiseuille Flow ◽  
Inertial Waves ◽  
Large Rotation ◽  
Rotation Numbers

The analogy between rotating shear flow and thermal convection suggests the existence of plumes, inertial waves and plume currents in plane Poiseuille flow under spanwise rotation. The existence of these flow structures is examined with the results of three-dimensional and two-dimensional three-component direct numerical simulations. The dynamics of plumes near the unstable side is embodied in a truncated exponential distribution of turbulent fluctuations. For large rotation numbers, inertial waves are identified near the stable side, and these can be used to explain the abnormal flow statistics, such as the large root-mean-square of the streamwise velocity fluctuation and the nearly negligible Reynolds shear stress. For small or medium rotation numbers, plumes generated from the unstable side form large-scale plume currents and the patterns of the plume currents show different capabilities in scalar transport.

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