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Register a new userUnified Flow and Thermal Law-of-the-wall Complete Formulations for Type-A Turbulent Boundary Layers
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In-depth analyses of existing direct numerical simulations (DNS) data from various sources supported a logical and important classification of generic turbulent boundary layers (TBL), namely Type-A, -B and -C TBL, based on distribution patterns of time-averaged wall-shear stress. Among these types, Type-A TBL and its related law, as represented by the DNS data of turbulence on a zero-pressure-gradient semi-infinite flat-plate, was investigated in terms of analytical formulations of velocity independent on Reynolds ( ) number. With reference to the analysis from von Karman in developing the conventional law-of-the-wall, the current study first physically distinguished the time-averaged local scale used by von Karman from the ensemble-averaged scale defined in the paper, and then derived the governing equations with the -independency under the ensemble-averaged scales. Based on indicator function (IDF) and TBL thickness, the sublayer partitions were rigorously defined. The analytical formulations for entire TBL, namely the complete law-of-the-wall, were established, including the formula in inner, buffer, semi-logarithmic (semi-log) and wake layer. The researches were featured by introducing the general damping and enhancing functions (GDF and GEF) and applying these functions to both linear and logarithmic coordinates. These law formulations were proved uniform and consistent in time-averaged local and ensemble-averaged scales, which were validated by the existing DNS and experiment data. Based on the similarity of relevant properly-scaled governing equations, the law formulations were logically reasoned being applicable to the temperature in Type-A thermal TBL. The findings advance the current understandings of the conventional TBL theory and its well-known foundations of law-of-the-wall.
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The direct measurement of wall shear stress in turbulent boundary layers (TBL) is challenging, therefore requiring it to be indirectly determined from mean profile measurements. Most popular methods assume the mean streamwise velocity to satisfy either a logarithmic law in the inner layer or a composite velocity profile with many tuned constants for the entire TBL, and require reliable data from the noise-prone inner layer. A simple method is proposed to determine the wall shear stress in zero pressure gradient TBL from measured mean profiles, without requiring noise-prone near-wall data. The method requires a single point measurement of mean streamwise velocity and mean shear stress in the outer layer, preferably between $20$ to $50$ $%$ of the TBL, and an estimate of boundary layer thickness and shape factor. The friction velocities obtained using the proposed method agree with reference values, to within $3$ $%$ over a range of Reynolds number.
We investigate a mechanism to manipulate wall-bounded flows whereby wave-like undulations of the wall topography drives the creation of bespoke longitudinal vortices. A resonant interaction between the ambient vorticity of the undisturbed shear flow and the undulation of streamlines enforced by the wall topography serves to slightly rotate the spanwise vorticity of the mean flow into the streamwise direction, creating a swirling motion, in the form of regular streamwise rolls. The process is kinematic and essentially identical to the `direct drive CL1 mechanism for Langmuir circulation (LC) proposed by Craik (1970). Boundary layers are modelled by selecting suitable primary flow profiles. A simple, easily integrable expression for the cross-plane stream function is found in two asymptotic regimes: the resonant onset of the essentially inviscid instability at early times, and the fully developed steady state viscous flow. Linear-order solutions for flow over undulating boundaries are obtained, fully analytical in the special case of a power-law profile. These solutions allow us to quickly map out the circulation response to boundary design parameters. The study is supplemented with direct numerical simulations which verify the manifestation of boundary induced Langmuir vortices in laminar flows with no-slip boundaries. Simulations show good qualitative agreement with theory. Quantitatively, the comparisons rest on a displacement length closure parameter adopted in the perturbation theory. While wall-driven LC appear to become unstable in turbulent flows, we propose that the mechanism can promote swirling motion in boundary layers, a flow feature which has been reported to reduce drag in some situations.
Wall cooling has substantial effects on the development of instabilities and transition processes in hypersonic boundary layers (HBLs). A sequence of linear stability theory, two-dimensional and non-linear three-dimensional DNSs is used to analyze Mach~6 boundary layers, with wall temperatures ranging from near-adiabatic to highly cooled conditions, where the second-mode instability radiates energy. Fluid-thermodynamic analysis shows that this radiation comprises both acoustic as well as vortical waves. 2D simulations show that the conventional trapped nature of second-mode instability is ruptured. Although the energy efflux of both acoustic and vortical components increases with wall-cooling, the destabilization effect is much stronger and no significant abatement of pressure perturbations is realized. In the near-adiabatic HBL, the wavepacket remains trapped within the boundary layer and attenuates outside the region of linear instability. However, wavepackets in the cooled-wall HBLs amplify and display nonlinear distortion, and transition more rapidly. The structure of the wavepacket displays different behavior; moderately-cooled walls show bifurcation into a leading turbulent head region and a trailing harmonic region, while highly-cooled wall cases display lower convection speeds and significant wavepacket elongation, with intermittent spurts of turbulence in the wake of the head region. This elongation effect is associated with a weakening of the lateral jet mechanism due to the breakdown of spanwise coherent structures. In moderately cooled-walls, the spatially-localized wall loading is due to coherent structures in the leading turbulent head region. In highly-cooled walls, the elongated near-wall streaks in the wake region of the wavepacket result in more than twice as large levels of skin friction and heat transfer over a sustained period of time.
The effect of rotation on the boundary layers (BLs) in a Rayleigh-Benard (RB) system at a relatively low Rayleigh number, i.e. $Ra = 4times10^7$, is studied for different Pr by direct numerical simulations and the results are compared with laminar BL theory. In this regime we find a smooth onset of the heat transfer enhancement as function of increasing rotation rate. We study this regime in detail and introduce a model based on the Grossmann-Lohse theory to describe the heat transfer enhancement as function of the rotation rate for this relatively low Ra number regime and weak background rotation $Rogtrsim 1$. The smooth onset of heat transfer enhancement observed here is in contrast to the sharp onset observed at larger $Ra gtrsim 10^8$ by Stevens {it{et al.}} [Phys. Rev. Lett. {bf{103}}, 024503, 2009], although only a small shift in the Ra-Ro-Pr phase space is involved.
We report an experimental study aiming to clarify the role of boundary conditions (BC) in high Rayleigh number $10^8 < {rm{Ra}} < 3 times 10^{12}$ turbulent thermal convection of cryogenic helium gas. We switch between BC closer to constant heat flux (CF) and constant temperature (CT) applied to the highly conducting bottom plate of the aspect ratio one cylindrical cell 30 cm in size, leading to dramatic changes in the temperature probability density function and in power spectral density of the temperature fluctuations measured at the bottom plate, while the dynamic thermal behaviour of the top plate and bulk convective flow remain unaffected. Within our experimental accuracy, we find no appreciable changes in Reynolds number Re(Ra) scaling, in the dimensionless heat transfer efficiency expressed via Nusselt number Nu(Ra) scaling, nor in the rate of direction reversals of large scale circulation.
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