# Interdisciplinary Applied Mathematics

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covers —7.5 < x < 7.5, —3.5 < y < 3.5, and —3.5 < z < 3.5. A sphere of radius R =1 is placed near the lower channel wall with its center at (xo,y0,z0) = (0, —1.5, 0). The fluid density and viscosity are chosen to be Pf = 1.0 and v = 1.0, respectively. Periodic conditions are imposed in the x-    and    z-directions,    and    no-slip    conditions    are applied    on    the two    walls

of the channel. The flow is driven by a constant force in the x-direction, F = jiwhere h is the channel half-width.

The spectral DLM simulations employ Fourier expansions in the x- and z-directions and a spectral element discretization in the y-direction (Dong et al., 2004). Distributed Lagrange multipliers are used to impose the zero-flow constraint inside the sphere. The “collocation points” consist of the flow grid points lying inside the sphere and intersection points between the surface of the sphere and the underlying flow grid lines. On the boundary

y    y

FIGURE 14.18. Sphere off center: comparison of velocity profiles between spectral DLM, DNS, and FCM at three streamwise locations at Reynolds number Re =1.55 (based on the channel center line maximum velocity and sphere diameter). Left: streamwise velocity. Right: wall-normal velocity.

collocation points the flow velocities are obtained via the spectral element interpolation procedure. In the FCM calculations a restoring force and torque are computed via a penalty method to keep the sphere from moving and rotating in the channel. The restoring force and torque are then used as the force monopole strength and dipole strength, respectively.

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