Determination of Reynolds shear stress from the turbulent mean velocity profile and spectral characteristics of Orr-Sommerfeld -Squire equations

Theoretically, based on a waveguide model, the expression of the tangential stress is formulated for steady, two-dimensional incompressible fluid flow over a flat plate in turbulent boundary layer. It is dependent on some factors, one of them, the behaviour of the last damping mode eigenvalues, and eigenfunctions, that are deduced from solution Orr-Sommerfeld equation by spectral Chebyshev collocation Method. Verification of the latter method is investigated by comparison the deduced formula of turbulent tangential stress with experimental data. In addition to, weight factors in this expression are connected to define the condition of dynamical system solution for multiple 3-wave resonance. This system is solved numerically, and the dynamic invariant is normalized to obtain the time average of the square modulus harmonic, and sub harmonics amplitudes by theorem Birkhoff-Khinchin. Comparison is made between the time-averaged and the phase average for the square modulus of harmonic, and sub harmonic amplitudes that defined on the unit sphere, in the state of multiple 3-wave resonance.

Shear Flow in Cylindrical Open Channel Under Precession

The study of forced oscillations in open cylindrical channel under precession is extended to include the shear effect, that is induced by inertial waves in such systems. The linear part of the problem led to two equations for stability one for the viscous part similar to Orr-Sommerfeld equation and one for the inviscid part similar to Rayleigh equation, the second was solved and discussed depending on the stream function observation. The linear part also led to relationship that connects between the stream velocity and the disturbance one is derived in a form similar to Burns conditions for open flows under normal conditions. Experimentally measuring the horizontal velocity distribution with depth showed that this distribution is sinusoidal one. Burns condition was solved based on this assumption. The nonlinear part of the problem led to a new version of Koteweg De-Vries (KdV) equation that is solved numerically by applying the leapfrog method for time discretization, Fourier transformation for the space one, and the trapezoidal rule for solving the integrals with depth, the results showed that the shear has no specific impact on the wave form which is similar to the classical results obtained by the theories under normal conditions.

Effect of latitudinal differential rotation on solar Rossby waves: Critical layers, eigenfunctions, and momentum fluxes in the equatorial β plane

Context. Retrograde-propagating waves of vertical vorticity with longitudinal wavenumbers between 3 and 15 have been observed on the Sun with a dispersion relation close to that of classical sectoral Rossby waves. The observed vorticity eigenfunctions are symmetric in latitude, peak at the equator, switch sign near 20°–30°, and decrease at higher latitudes. Aims. We search for an explanation that takes solar latitudinal differential rotation into account. Methods. In the equatorial β plane, we studied the propagation of linear Rossby waves (phase speed c < 0) in a parabolic zonal shear flow, U = − U̅ ξ2 < 0, where U̅ = 244 m s−1, and ξ is the sine of latitude. Results. In the inviscid case, the eigenvalue spectrum is real and continuous, and the velocity stream functions are singular at the critical latitudes where U = c. We add eddy viscosity to the problem to account for wave attenuation. In the viscous case, the stream functions solve a fourth-order modified Orr-Sommerfeld equation. Eigenvalues are complex and discrete. For reasonable values of the eddy viscosity corresponding to supergranular scales and above (Reynolds number 100 ≤ Re ≤ 700), all modes are stable. At fixed longitudinal wavenumber, the least damped mode is a symmetric mode whose real frequency is close to that of the classical Rossby mode, which we call the R mode. For Re ≈ 300, the attenuation and the real part of the eigenfunction is in qualitative agreement with the observations (unlike the imaginary part of the eigenfunction, which has a larger amplitude in the model). Conclusions. Each longitudinal wavenumber is associated with a latitudinally symmetric R mode trapped at low latitudes by solar differential rotation. In the viscous model, R modes transport significant angular momentum from the dissipation layers toward the equator.

Spatial heterogeneity in subglacial drainage driven by till erosion

The distribution and drainage of meltwater at the base of glaciers sensitively affects fast ice flow. Previous studies suggest that thin meltwater films between the overlying ice and a hard-rock bed channelize into efficient drainage elements by melting the overlying ice. However, these studies do not account for the presence of soft deformable sediment observed underneath many West Antarctic ice streams, and the inextricable coupling that sediment exhibits with meltwater drainage. Our work presents an alternate mechanism for initiating drainage elements such as canals where meltwater films grow by eroding the sediment beneath. We conduct a linearized stability analysis on a meltwater film flowing over an erodible bed. We solve the Orr–Sommerfeld equation for the film flow, and we compute bed evolution with the Exner equation. We identify a regime where the coupled dynamics of hydrology and sediment transport drives a morphological instability that generates spatial heterogeneity at the bed. We show that this film instability operates at much faster time scales than the classical thermal instability proposed by Walder. We discuss the physics of the instability using the framework of ripple formation on erodible beds.

Calculation of the Orr-Sommerfeld stability equation for the plane Poiseuille flow

The stability of plane Poiseuille flow depends on eigenvalues and solutions which are generated by solving Orr-Sommerfeld equation with input parameters including real wavenumber and Reynolds number . In the reseach of this paper, the Orr-Sommerfeld equation for the plane Poiseuille flow was solved numerically by improving the Chebyshev collocation method so that the solution of the Orr-Sommerfeld equation could be approximated even and odd polynomial by relying on results of proposition 3.1 that is proved in detail in section 2. The results obtained by this method were more economical than the modified Chebyshev collocation if the comparison could be done in the same accuracy, the same collocation points to find the most unstable eigenvalue. Specifically, the present method needs 49 nodes and only takes 0.0011s to create eigenvalue while the modified Chebyshev collocation also uses 49 nodes but takes 0.0045s to generate eigenvalue with the same accuracy to eight digits after the decimal point in the comparison with , see [4], exact to eleven digits after the decimal point.

On the large difference between Benjamin’s and Hanratty’s formulations of perturbed flow over uneven terrain

Flow over an uneven terrain is a complex phenomenon that requires a chain of approximations in order to be studied. In addition to modelling the intricacies of turbulence if present, the problem is classically first linearized about a flat bottom and a locally parallel flow, and then asymptotically approximated into an interactive representation that couples a boundary layer and an irrotational region through an intermediate inviscid but rotational layer. The first of these steps produces a stationary Orr–Sommerfeld equation; since this is a one-dimensional problem comparatively easy for any computer, it would seem appropriate today to forgo the second sweep of approximation and solve the Orr–Sommerfeld problem numerically. However, the results are inconsistent! It appears that the asymptotic approximation tacitly restores some of the original problem’s non-parallelism. In order to provide consistent results, Benjamin’s version of the Orr–Sommerfeld equation needs to be modified into Hanratty’s. The large difference between Benjamin’s and Hanratty’s formulations, which arises in some wavenumber ranges but not in others, is here explained through an asymptotic analysis based on the concept of admittance and on the symmetry transformations of the boundary layer. A compact and accurate analytical formula is provided for the wavenumber range of maximum laminar shear-stress response. We highlight that the maximum turbulent shear-stress response occurs in the quasi-laminar regime at a Reynolds-independent wavenumber, contrary to the maximum laminar shear-stress response whose wavenumber scales with a power of the boundary-layer thickness. A numerical computation involving an eddy-viscosity model provides a warning against the inaccuracy of such a model. We emphasize that the range $k\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}<10^{-3}$ of the spectrum remains essentially unexplored, and that the question is still open whether a fully developed turbulent regime, similar to the one predicted by an eddy-viscosity model, ever exists for open flow even in the limit of infinite wavelength.