High-resolution 3D computation of time-periodic long-wake flows with the Carrier-Domain Method and Space–Time Variational Multiscale method with isogeometric discretization

Abstract

The Carrier-Domain Method was introduced for high-resolution computation of time-periodic long-wake flows. The cost-effectiveness of the method makes such computations practical in 3D. A short segment of the wake domain, the carrier domain, moves in the free-stream direction, from the beginning of the long wake domain to the end. The data at the moving inflow plane comes from the time-periodic data computed at an earlier position of the carrier domain. With the high mesh resolution that can easily be afforded over the short domain segment, the wake flow patterns can be carried, with superior accuracy, far downstream. Computing the long-wake flow with a high-resolution moving mesh that covers a short segment of the wake domain at any instant during the computation would certainly be far more cost-effective than computing it with a high-resolution fixed mesh that covers the entire length. We present high-resolution 3D computation of time-periodic long-wake flow for a cylinder and a wind turbine, both computed with isogeometric discretization and the Space–Time Variational Multiscale method. In the isogeometric discretization, the basis functions are quadratic NURBS in space and linear in time. The cylinder flow is at Reynolds number 100. At this Reynolds number, the flow has an easily discernible vortex shedding period. The wake flow is computed up to 350 diameters downstream of the cylinder, far enough to see the secondary vortex street. In the wind turbine long-wake flow computation, the velocity data at the inflow boundary of the wake domain comes from an earlier wind turbine computation, with the turbine rotor having a diameter of ${126}\,\hbox {m}$, extracted by projection from a plane located ${10}\,\hbox {m}$ downstream of the turbine. The wake flow is computed up to ${482}\,\hbox {m}$ downstream of the wind turbine. In both the cylinder and wind turbine wake flow computations, the flow patterns obtained with the full domain and carrier domain show a near-perfect match, clearly demonstrating the effectiveness and practicality of the Carrier-Domain Method in high-resolution 3D computation of time-periodic long-wake flows.

Appendix A: Stabilization parameters and element lengths

The stabilization parameter $\tau _{\textrm{SUPS}}$ is mostly from [29]:

$$\begin{aligned} \tau _{\textrm{SUPS}}&= \left( \tau _{\textrm{SUGN12}}^{-2} + \tau _{\textrm{SUGN3}}^{-2} + \tau _{\textrm{SUGN4}}^{-2}\right) ^{-\frac{1}{2}}. \end{aligned}$$

(2)

The first two components are given as

$$\begin{aligned} \tau _{\textrm{SUGN12}}^{-2}&= \begin{bmatrix} 1 \\ \textbf{u}\\ \end{bmatrix} \begin{bmatrix} 1 \\ \textbf{u}\\ \end{bmatrix} : \textbf{G}^\textrm{ST} \end{aligned}$$

(3)

and

$$\begin{aligned} \tau _{\textrm{SUGN3}}^{-1}&= \nu \textbf{r}\textbf{r}:\textbf{G}, \end{aligned}$$

(4)

where $\textbf{r}$ is the solution-gradient direction:

$$\begin{aligned} \textbf{r}&= \frac{\pmb {\nabla }\left\| \textbf{u}\right\| }{\left\| \pmb {\nabla }\left\| \textbf{u}\right\| \right\| }. \end{aligned}$$

(5)

Here $\textbf{G}$ and $\textbf{G}^\textrm{ST}$ are the space-only and ST element metric tensors:

(6)

(7)

where

$$\begin{aligned} \hat{\textbf{Q}}= & {} \textbf{Q} \cdot \textbf{D}^{-1}, \end{aligned}$$

(8)

$$\begin{aligned} \hat{\textbf{Q}}^\textrm{ST}= & {} \textbf{Q}^\textrm{ST} \cdot \left( \textbf{D}^\textrm{ST}\right) ^{-1}. \end{aligned}$$

(9)

The space-only and ST Jacobian tensors are

$$\begin{aligned} \textbf{Q}= & {} \frac{\partial \textbf{x}}{\partial \pmb {\xi }}, \end{aligned}$$

(10)

$$\begin{aligned} \textbf{Q}^\textrm{ST}= & {} \begin{bmatrix} \frac{\partial t}{\partial \theta } &{} \frac{\partial t}{\partial \pmb {\xi }} \\ \frac{\partial \textbf{x}}{\partial \theta } &{} \textbf{Q} \\ \end{bmatrix} , \end{aligned}$$

(11)

and $\textbf{D}$ and $\textbf{D}^\textrm{ST}$ represent the transformation tensors relating the integration and “preferred” parametric spaces (see [10]). The tensor $\textbf{D}^\textrm{ST}$ is given as

(12)

With that, we can express Eqs. (6) and (7) as

(13)

(14)

The definitions used for $D_\theta$ and $\textbf{D}$ play an important role, especially in higher-order IGA discretization [9, 10] and simplex elements [134].

The third component, originating from [6], is defined as

$$\begin{aligned} \tau _{\textrm{SUGN4}}= \left\| \pmb {\nabla }\textbf{u}^h\right\| _F^{-1} , \end{aligned}$$

(15)

where $\Vert \cdot \Vert _F$ represents the Frobenius norm.

The stabilization parameter $\nu _{\textrm{LSIC}}$ is from [138]:

$$\begin{aligned} \nu _{\textrm{LSIC}}&= \frac{h^2_\textrm{LSIC}}{\tau _{\textrm{SUPS}}}, \end{aligned}$$

(16)

where the element length $h_\textrm{LSIC}$ is typically set equal to the minimum element length:

$$\begin{aligned} h_\textrm{MIN} = 2 \left( \max _{\textbf{r}} \left( \textbf{r} \textbf{r} : \textbf{G} \right) \right) ^{-\frac{1}{2}} . \end{aligned}$$

(17)