HiFi-TURB-DLR rounded step

Semi-confined flows

Underlying Flow Regime 3-36

Test Case Study

Brief Description of the Study Test Case

The geometry of this UFR alongside the mesh is shown in Fig. 1. The geometry comprises three main sections: Constant-Width Forebody section Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {L_F}} with the largest width Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {Y_F}} , Contoured Boat-tail section Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {L_B}} with the contoured width Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {Y_B}} and Constant-Width-Aftbody section Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {L_A}} with the smallest width Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {Y_A}} . The width of the last section is modified to generate the desired APG. This modification is achieved through the variation of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\sigma}} , which is the ratio of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {Y_A}} to Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {Y_F}} .

Figure 1: Flow Domain and grid of RANS simulations

The parametric geometry definition for the three relevant sections is given in [‌6] and is depicted in Fig. 1. The axial origin Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {x=0}} is set at the beginning of the Boattail section.

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle { \begin{align} y_{wall} &= y_F = H/(1-\sigma) \qquad \mathrm{for} \qquad -L_F < x/H \leq 0 \\ y_{wall} &= y_B = a_1 + a_2 x^3 + a_3 x^4 + a_4 x^5 \qquad \mathrm{for} \qquad 0 < x/H \leq L_B \qquad\qquad (1) \\ y_{wall} &= Y_A = \sigma H/(1 - \sigma) \qquad \mathrm{for} \qquad L_B < x/H \leq L_B + L_A \\ \\ \end{align} }}

with Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {a_1 = H/(1-\sigma)}} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {a_2 = - 10 H/L_B^3}} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {a_3 = 15 H/L_B^4}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {a_4 = - 6 H/L_B^5}} .

CFD Methods

Reynolds-Averaged Navier-Stokes computations

For the entire computational domain, a structured 2D mesh was created using Pointwise V18.2. Sensitivity studies were carried out. The final mesh contains a total of 266.112 points. Along the body contour 448 points are used in streamwise direction with a smaller spacing in the focus region. In the normal direction to the body wall 298 points are used, 98 of which are concentrated near the body wall region. The body wall-normal growth ratio is approximatively 1.077 and the dimensionless distance from the wall is Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {y^+<1}} along the body wall for all meshes and simulation scenarios. For the inflow boundary situated at Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {x = -4.5 R_{max}}} a reservoir-pressure inflow boundary condition is used. This boundary condition prescribes total pressure and total density. The inflow direction is by default perpendicular to the boundary face. For the outflow boundary at Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {x = 7.6 {max}}} an exit-pressure outflow boundary condition is used. The exit pressure is adapted during the simulation to match the reference pressure at the coordinate point Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {Z_{ref} = (-0.15m,0m,0.18m)}} . The upper boundary is a far-field boundary condition situated Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {68 R_{max}}} from the viscous body wall. Symmetry boundary condition is used on both side planes of the 2D domain. Different Reynolds numbers were simulated, two of them are presented in Table 2 with the corresponding reference parameters. Here we use a definition of the Reynolds number based on Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {R_{max}}} as Reynolds length.

DNS Computations

The test case is designed as a numerical experiment with the aim of comparing RANS results to DNS data. For the set-up of the DNS, the inflow boundary conditions are different, e.g. a recycling method can be used to generate the turbulent input or synthetic turbulence can be injected. It is also possible to numerically trip the boundary layer from laminar to turbulent to generate the desired turbulent boundary layer. Hence, to ensure a comparison to the results achieved with RANS turbulence models, a reference position upstream of the APG-area is defined where boundary-layer properties need to match between RANS and DNS computations to permit the comparison downstream in the region of interest. The reference position is located at Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {x/H = -3.51}} . Depending on the generation of turbulence at the inlet, the computational domain needs to be adapted to ensure the correct boundary-layer properties at the reference position. If numerical tripping is performed, the laminar and turbulent distances need to be determined upstream of the reference position by precursor simulation as displayed in Fig. 3.

Figure 3: Set-up for DNS Simulation

At the reference position, the properties of the turbulent boundary layer are determined by the Reynolds number based on the momentum thickness Reθ and the Reynolds number based on the friction velocity Reτ. For the incipient and the moderate separation case, for the SA-neg model with RC and QCR, with and without LowRe-Modification, the numbers are given in Table 3.

Between the different flow condition proposed, UNIBG focused the effort on the incipient separation configuration case with Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {Re_H = 78,490}} . A precursory computational campaign for the turbulent flow over a flat plate has been performed with the purpose of (i) assessing the effectiveness of the Synthetic Inlet Turbulence (SIT) injection strategy inspired by the work of Housseini et al. [‌18] and Schlatter and Örlü [‌19]; (ii) investigating the influence of the mesh density on the solution; (iii) defining the inlet boundary position and flow condition that ensure the target boundary layer integral parameters at the reference location Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {x_{ref}}} (see Figure 3). According to the outcomes of this campaign, the inlet boundary is set at Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {x/H = -12.71}} , where the Blasius laminar velocity profile computed at Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {Re_x=650,000}} , the uniform static pressure and the uniform total temperature (see Table 2) are imposed. At location Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {x/H = -12.1}} , within the laminar boundary layer region, the tripping source term is activated to promote the transition to turbulence [‌18][‌19]. The outlet boundary is positioned at Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {x/H = 23.95}} , with a pressure outflow condition having the same value of the inlet static pressure and at the end of a streamwise coarsened mesh region starting at Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {x/H = 13.82}} . Following the setup of the RANS computations, the no-slip adiabatic boundary condition is set while at the upper boundary, situated at Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {179 H}} from the wall upstream the smooth bump, the far-field boundary condition is imposed. Side planes, instead, are considered as periodic with a distance from each other of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\Delta z = 3 H}} . The DNS have been performed by using the UNIBG in-house software MIGALE [‌20]. MIGALE is an implicit high-order discontinuous Galerkin solver for the compressible and incompressible Navier-Stokes equations. Godunov fluxes are treated with the exact solution of the local Riemann problems while viscous fluxes are handled by means of the BR2 scheme [‌21]. The time integration is performed with linearly implicit Rosenbrock type Runge-Kutta schemes with optimal stability properties up to order five. The computational mesh is made of 15,016,384 hexahedral elements with quadratic edges concentrated near the wall region. The wall-normal growth ratio is approximatively 1.2 with a first cell height of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {y^+ ~ 1}} . Time integration is performed with the fifth order – eight stages Rosenbrock scheme ROD5_1 [‌22] using a global time step adaptation strategy [‌23]. The corresponding average step size is Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\Delta t = 19/14000}} CTU, where the convective time unit (CTU) is defined as the ratio between Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {H}} and the freestream velocity. Turbulence statistics have been collected for 26 CTU.