Topic C — Transient Vertical Bed with Buoyancy

Topic B made time-dependent · Boussinesq buoyancy in a Brinkman flow · two-way coupling · tier: advanced (build B first)

Motivation

Topic B asks whether a steady operating point exists. The safety question is dynamic: starting cold, does the bed settle or run away? And when the bed heats, buoyancy changes the flow, which changes residence time and cooling — reaction feeds back into the flow field. This is the only topic with two-way flow–temperature coupling.

Course weeks: topic B, plus 10 (θ-method, stability), 11 (Taylor–Hood), 14 (splitting schemes). New: Boussinesq coupling, a segregated iteration inside each time step.

Model

Vertical bed, cooled side walls, buoyant upflow and the segregated coupling in each time step (schematic).

Vertical bed x\in[0,1] (cooled side walls), y\in[0,4] (flow upward from y=0), gravity -\hat{\mathbf e}_y.

Transient Brinkman–Boussinesq flow: \partial_t\mathbf u-\nabla\cdot(\nu\nabla\mathbf u)+\frac\nu\kappa\mathbf u+\nabla p=\gamma\,(T-T_\text{ref})\,\hat{\mathbf e}_y,\qquad\nabla\cdot\mathbf u=0, \gamma=\rho_0\beta g (lumped buoyancy strength). Inlet velocity at y=0, no-slip side walls, do-nothing outlet.

Species and energy as in topic B with time derivatives: \varepsilon\,\partial_tc+\mathbf u\cdot\nabla c-\nabla\cdot(D\nabla c)=-R,\qquad\partial_tT+\mathbf u\cdot\nabla T-\nabla\cdot(k\nabla T)=\Delta H\,R,\qquad R=k_0e^{-E/T}c, Newton cooling on x=0,1; cold, unreacted initial state.

Time stepping (required baseline): backward Euler; in each step a segregated (Picard) iteration — (1) flow with the current T, linear; (2) coupled (c,T) with that \mathbf u, Newton — repeated until T changes by less than a tolerance.

Design output: T_{\max}(t) and \max|\mathbf u|(t) for several T_\text{in}, with and without buoyancy (\gamma=0); classification settles vs runs away.

Note

Analysis questions (required). (i) Stability of backward Euler for the linearised energy equation (Week 10) — and why that does not exclude a physical runaway. (ii) When does the segregated iteration converge? Estimate its contraction factor from your runs and relate it to \gamma. (iii) Inf-sup stability of your flow pair with the Brinkman term (Week 11).

Required tasks

  1. Steady Brinkman–Boussinesq flow with a prescribed warm patch: warm fluid rises (sign check).
  2. Transient (c,T) without buoyancy; run to steady state.
  3. Segregated coupling with buoyancy, increasing \gamma from 0; the T_\text{in} sweep with and without buoyancy. Does buoyancy help or hurt in upflow? Decide from your results.
  4. Time-step study (\Delta t halved twice).

Verification specific to this topic

  • Steady cross-check: with \gamma=0 the long-time limit must equal the steady solution of the same model computed directly (your own steady Newton solver at the same parameters, i.e. topic B rotated). Report both values and the difference.
  • Transient energy balance: storage + convection + wall loss = generation, integrated over the domain and over time.
  • MMS in time: a manufactured solution \propto e^{-t} (or \sin t) for the energy equation — first-order temporal rate for backward Euler.
  • Spatial MMS for the flow with the Boussinesq term (Taylor–Hood rates).

Stretch goals

  1. Monolithic (\mathbf u,p,c,T) Newton instead of the segregated iteration; cost and robustness compared.
  2. Downflow: buoyancy opposes the flow — recirculation near the hot spot?
  3. Adaptive \Delta t (halve on failure).
  4. Critical buoyancy strength at fixed T_\text{in}.

Milestones

M1 flow with a prescribed temperature · M2 transient (c,T), \gamma=0, steady cross-check passed · M3 segregated coupling, small \gamma · M4 sweeps, time-step study · M5 stretch goal. Do not pass M2 before the steady cross-check agrees.

Pitfalls

  • “Runaway” must be distinguished from a numerical blow-up: halve \Delta t — a physical runaway stays, an instability moves.
  • A cold start at a high k_0 needs small first steps.

References

  • D. A. Nield, A. Bejan, Convection in Porous Media, Springer — Boussinesq–Brinkman models.
  • J. Donea, A. Huerta, Finite Element Methods for Flow Problems, Wiley 2003 — time integration of coupled flow problems.