Let \(\Omega \in \mathbb {R}^2\) be the computational domain. Incompressible, viscous fluid motion in spatial domain Ω over a time interval (0, T) is governed by the incompressible Navier-Stokes equations with vector-valued velocity u, scalar-valued pressure p, kinematic viscosity ν and a body forcing f:
$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} &\displaystyle =&\displaystyle - \nabla p + \nu \Delta \mathbf{u} + \mathbf{f}, {} \end{array} \end{aligned} $$
(1)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \nabla \cdot \mathbf{u} &\displaystyle =&\displaystyle 0. {} \end{array} \end{aligned} $$
(2)
Boundary and initial conditions are prescribed as
$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathbf{u} &\displaystyle =&\displaystyle \mathbf{d} \quad \text{ on } \Gamma_D \times (0, T), \end{array} \end{aligned} $$
(3)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \nabla \mathbf{u} \cdot \mathbf{n} &\displaystyle =&\displaystyle \mathbf{g} \quad \text{ on } \Gamma_N \times (0, T), \end{array} \end{aligned} $$
(4)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathbf{u} &\displaystyle =&\displaystyle {\mathbf{u}}_0 \quad \text{ in } \Omega \times 0, {} \end{array} \end{aligned} $$
(5)
with d, g and u
0 given and ∂ Ω = ΓD ∪ ΓN, ΓD ∩ ΓN = ∅. The Reynolds number Re, which characterizes the flow [11], depends on ν, a characteristic velocity U, and a characteristic length L:
$$\displaystyle \begin{aligned} Re = \frac{UL}{\nu}. \end{aligned} $$
(6)
We are interested in computing the steady states, i.e., solutions where \(\frac {\partial \mathbf {u}}{\partial t}\) vanishes. The high-order simulations are obtained through time-advancement, while the ROM solutions are obtained with a fixed-point iteration.
2.1 Oseen-Iteration
The Oseen-iteration is a secant modulus fixed-point iteration, which in general exhibits a linear rate of convergence [2]. Given a current iterate (or initial condition) u
k, the next iterate u
k+1 is found by solving linear system:
$$\displaystyle \begin{aligned} \begin{array}{rcl} -\nu \Delta {\mathbf{u}}^{k+1} + ({\mathbf{u}}^k \cdot \nabla) {\mathbf{u}}^{k+1} + \nabla p &\displaystyle =&\displaystyle \mathbf{f} \text{ in } \Omega, {} \\ \nabla \cdot {\mathbf{u}}^{k+1} &\displaystyle =&\displaystyle 0 \text{ in } \Omega, \\ {\mathbf{u}}^{k+1} &\displaystyle =&\displaystyle \mathbf{d} \quad \text{ on } \Gamma_D, \\ \nabla {\mathbf{u}}^{k+1} \cdot \mathbf{n} &\displaystyle =&\displaystyle \mathbf{g} \quad \text{ on } \Gamma_N. \end{array} \end{aligned} $$
Iterations are typical stopped when the relative difference between iterates falls below a predefined tolerance in a suitable norm, like the L
2( Ω) or \(H^1_0(\Omega )\) norm.
2.2 Model Description
We consider the reference computational domain shown in Fig. 1, which is decomposed into 36 triangular spectral elements. The spectral element expansion uses modal Legendre polynomials of the Koornwinder-Dubiner type of order p = 11 for the velocity. Details on the discretization method can be found in chapter 3.2 of [12]. The pressure ansatz space is chosen of order p − 2 to fulfill the inf-sup stability condition [1, 20]. A parabolic inflow profile is prescribed at the inlet (i.e., x = 0) with horizontal velocity component u
x(0, y) = y(3 − y) for y ∈ [0, 3]. At the outlet (i.e., x = 8) we impose a stress-free boundary condition, everywhere else we prescribe a no-slip condition.
The height of the narrowing in the reference configuration is μ = 1, from y = 1 to y = 2. See Fig. 1. Parameter μ is considered variable in the interval μ ∈ [0.1, 2.9]. The narrowing is shrunken or expanded as to maintain the geometry symmetric about line y = 1.5. Figures 2, 3, and 4 show the velocity components close to the steady state for μ = 1, 0.1, 2.9, respectively.
The viscosity is kept constant to ν = 1. For these simulations, the Reynolds number (6) is between 5 and 10, with maximum velocity in the narrowing as characteristic velocity U and the height of the narrowing characteristic length L. For larger Reynolds numbers (about 30), a supercritical pitchfork bifurcation occurs giving rise to the so-called Coanda effect [8, 9, 22], which is not subject of the current study. Our model is similar to the model considered in [18], i.e. an expansion channel with an inflow profile of varying height. However, in [18] the computational domain itself does not change.