Triple-deck analysis of the steady flow over a rotating disk with surface roughness

The effect of surface roughness on the steady laminar flow induced by a rotating disk submerged by fluid otherwise at rest is investigated here theoretically and numerically. A theory is proposed where a triple-deck analysis is applied leading to a fast evaluation of the steady-flow modification due to the rough surface. The theory assumes that the roughness is much smaller than the boundary-layer height and is characterized by a significantly longer length scale (slender roughness). Only the leading-order correction is developed here, corresponding to a velocity-field correction that is linear with the roughness height. The proposed theory neglects some curvature terms (here partially accounted by means of a stretching of the radial coordinate and of a scaling of the dependent variables). Numerical simulations performed with different roughness geometries (axisymmetric roughness, radial grooves, and localized bumps) have been used to validate the theory. Results indicate that the proposed theory leads to a good quantification of the flow modifications due to surface roughness at a very low computational cost. A demonstration of the capabilities of the theory is finally proposed where the statistical effects on the flow due to a random (but statistically known) roughness distributed on the surface of a rotating disk are characterized.

Figure 1

Self-similar solution of von Kármán flow in the inertial reference frame (left) and three-dimensional representation of the resulting velocity field (right).

Figure 2

Schematic description of the triple-deck structure. The thick arrows refer to matching boundary conditions between the various layers.

Figure 3

Computational domain on freefem$++$ for the considered axisymmetric groove. The dashed line represents the symmetry axis. Capital letters indicate the applied boundary conditions: (V) von Kármán self-similar solution, (O) stress-free condition, (W) no-slip condition. In the lower part of the figure a closeup of the roughness distribution is shown.

Figure 4

Computational domain projected on the $r\text{−}z$ (top) and $\theta \text{−}z$ plane. Capital letters indicate the applied boundary conditions: (V) von Kármán velocity solution, (O) stress-free condition, (W) no-slip condition, and (P) periodic (cyclic) condition. The roughness is represented on the $(r,z)$ and ($\theta ,z$) plane for the Gaussian bump ($a$) and groove ($b$), respectively.

Figure 5

Maximum azimuthal (filled symbols) and radial (empty symbols) velocity variation due to a bump with different values $h_{\max }^{+}$ and ${\sigma }_r$ from the available numerical simulations. ($*$) ${\sigma }_r=0.5$, ($\circ$) ${\sigma }_r=1$, ($+$) ${\sigma }_r=2$, and ($▿$) ${\sigma }_r=4$. The dotted line indicates a general linear relationship.

Figure 6

Azimuthal velocity variation due to an axisymmetric groove with $h_{\max }^{+}=0.1$ and $\lambda =4$ (top) or $\lambda =10$ (bottom) from the DNS and the TDT.

Figure 7

Radial (top) and azimuthal (bottom) components of the perturbation for the vertical plane along the isolated bump with ${\sigma }_r=4$ from the DNS and the TDT.

Figure 8

Comparison of velocity perturbation evaluated by TDT and DNS: radial (top), azimuthal (middle), and wall-normal (bottom) component at five different positions ($r^{+}=250,\theta$). From left to right $\theta ={[44.1,45,45.7,46.4,46.9]}^{\circ }$. The axis values are fixed for each component, thus they are reported only for the first column. The gray vertical line highlights the zero value on the $x$ axis for each position.

Figure 9

Radial (top) and azimuthal (bottom) components of the perturbation for the horizontal plane at $z^{+}=0.2$ for the isolated 3D bump with ${\sigma }_r=4$ from the DNS and the TDT.

Figure 10

Pressure correction for the horizontal plane at $z^{+}=0.2$ for the isolated 3D bump with ${\sigma }_r=4$ from the DNS and the TDT.

Figure 11

Radial (top) and azimuthal (bottom) components of the perturbation for the vertical plane along the radial groove from the DNS and the TDT.

Figure 12

Radial (top) and azimuthal (bottom) components of the perturbation for the horizontal plane at $z^{+}=0.2$ for the radial groove from the DNS and the TDT.

Figure 13

Summary of numerical setup for gPC procedure. (Top) Definition of stochastic parameters. (Bottom) Tensor product grid and probability density function of parameters.

Figure 14

Mean value of radial (top) and azimuthal (bottom) velocity components evaluated with gPC approach for sinusoidal roughness with fixed height $h_{\max }^{+}=0.1$ from the DNS and the TDT. The black dashed lines indicate the corresponding mean value of the perturbation velocity at $r^{+}=250$.

Figure 15

Standard deviation of radial (top) and azimuthal (bottom) velocity components evaluated with gPC approach for sinusoidal roughness with fixed height $h_{\max }^{+}=0.1$ from the DNS the TDT. The black dashed lines indicate the corresponding standard deviation of perturbation velocity at $r^{+}=250$.

Figure 16

Azimuthal velocity variation due to an axisymmetric bump with $h_{\max }^{+}=0.1$ and ${\sigma }_r=2$ (top) or ${\sigma }_r=4$ (middle) from the DNS and the TDT. (Bottom) azimuthal field for $h_{\max }^{+}=0.8$ and ${\sigma }_r=4$.