Boundary conditions for thermally coupled flows

can be seen that the Prandtl number is a measure for the similarity of the transport of heat and momentum. The Grashof number is the ratio of the buoyancy force to the viscous force.

Depending on the boundary conditions for the momentum equation, there are two different possibilities for choosing a characteristic scaling velocity ˜Usc. In the case of so-calledforced convection, the fluid motion is enforced by the boundary conditions (see section 1.2). Then we choose ˜U_sc=||u||∞,Γ. In indoor-air flow problems most of the time there is no external force and u = 0 or a homogeneous Neumann condition is prescribed on the boundary.

The only driving forces are due to buoyancy effects. Then physically meaningful choice is ˜U_sc = ( ˜β₀|g˜|T˜_{dif f}L)˜ ^1/2, cf. [KC93], p.408. In both cases the reduced pressure is non-dimensionalised with ˜ρ₀U˜_sc².

Remark 1.2

As it will turn out in Section 10.5, an appropriate choice for ˜U_sc is essential for the PSPG-stabilisation technique in the numerical solution process. ♦ In this thesis dimensionless quantities are chosen in agreement with [Mue99], viz.,

˜ a≡ λ˜

˜

c_pρ˜₀ , a≡ ˜a

L˜U˜_sc, g≡ g˜L˜

U˜_sc² , cp ≡ λ˜T˜_{dif f}

˜

ρ₀˜aU˜_sc² , q˙^V ≡ q˜˙^VL˜

˜

ρ₀U˜_sc³ , β ≡β˜0T˜_{dif f} , ν≡ µ˜

˜ ρ₀U˜_scL˜. This yields the following system of equations

∂tu−∇· ( 2νS(u) ) + (u·∇)u+∇p_red = −β θg , (1.9)

∇·u = 0 , (1.10)

∂tθ+ (u·∇)θ−∇·(a∇θ) = q˙^Vc⁻¹_p . (1.11)

1.2. Boundary conditions for thermally coupled flows

For specifying the boundary conditions, we introduce two partitions of Γ : one for the momentum equation and one for all scalar equations, e.g., the heat transfer equation and a possible additional equation describing contaminant transport.

The first partition of Γ is due to the boundary conditions concerning the momentum equa-tion. For this purpose we define thestress tensor

σ(u, p) = −pI + 2νS(u) .

Moreover we suppose that for almost every pointxin Γ we have a local orthonormal basis {n(x) , t_j(x) ,1≤j≤d−1}, where {t_j}^d_j=1⁻¹ is a local orthonormal basis for the tangent space of Γ inxand ndenotes the outer unit normal vector to Γ atx. Denote

ΓF = {x ∈ Γ | u = uF , uF ·n<0 a.e. in ΓF }, (1.12)

ΓW = {x ∈ Γ | u·n = 0, χn^Tσ(u, p)tj = σt(u)·tj 1≤j≤d−1 } , (1.13)

Γ_N = {x ∈ Γ | σ(u, p)n = σ_n} (1.14)

1. The laminar model

which are mutually disjoint and satisfy Γ_F ∪Γ_N ∪Γ_W = Γ. The quantityn^TT r|ΓWσ(u, p) is called stress vector, which represents the force that the fluid exerts on the wall. Here T r|Γ_W denotes the trace operator, see Chapter B and Remark 8.4. ΓF is a forced convection inflow boundary; on Γ_F a non-zero inflow velocity profile is prescribed. (1.13) describes a general (non-linear) friction law, covering the following situations:

(i) slip with linear friction: χ≡1, andσt(u)·tj ≡ −βju·tj, (ii) wall stress condition: χ≡1, andσt(u)·tj ≡τw u·tj

||u·tj|| (provided u·tj 6= 0), (iii) no-slipcondition: χ= 0, andσt(u)·tj ≡ −u·tj.

Note that in the case of(i),σt(u)·tj depends linearly on the magnitude of u·tj whereas in the case of (ii), only a directional and a so-called phase information of u·t_j is used.

Due to the definition of Γ_F, even in case(iii) Γ_F and Γ_W are disjoint.

Now we explain how different physical situations can be modelled using these types of boundary conditions. Informally spoken, in indoor-air flow simulations the boundary con-sists of openings and solid impermeable and smooth walls. On the wall, in any case we impose u·n = 0, being covered by (1.13), (iii). Next openings are studied. There is a wide agreement that σ(u, p)n =0 is suitable to model undisturbed outflow. Concerning inflow, we have to distinguish between forced convection and natural convection. In the former case, on a part of the boundary a nonzero inflow velocity is prescribed, i.e. Γ_F 6=∅. Alternatively, inflow can be enforced by imposing a suitable external pressureσ_n in (1.14).

Of course, when selecting (1.14), it is possible that u=0 or u·n= 0 on parts of ΓN. In the latter case of natural convection, i.e. Γ_F = ∅, σ_n = 0 in (1.14), the fluid motion is induced by buoyancy forces. It is worth rewriting both cases in the following form:

Forced convection: ΓF 6=∅ orσn6=0.

Natural convection: Γ_F =∅ andσn=0.

In most indoor-air flow problems both natural and forced convection have to be considered.

This case is also referred to as mixed convection. As pointed out in [KC93], in mixed convection problems often the forced convection character dominates, in particular if Gr is small compared to Re. The crucial question is whether the buoyancy force term in the momentum equation is significant or not.

The most general condition describing solid impermeable walls is (1.13). Measurements showed that no-slip is the correct boundary condition on walls for indoor-air flow problems, cf. [Nei99]. However, as it will turn out later, it is useful considering the more general condition (1.13).

A second partition of Γ can be defined w.r.t. the sign of u·n, wherendenotes the outer unit vector normal to Γ, viz.,

1.2. Boundary conditions for thermally coupled flows

Γ₋(u) = { x ∈ Γ | u·n < 0} inflow boundary, (1.15)

Γ0(u) = { x ∈ Γ | u·n = 0} ”wall” except a set of measure zero, (1.16)

Γ₊(u) = { x ∈ Γ | u·n > 0} outflow, (1.17)

which are mutually disjoint and satisfy Γ₋(u)∪Γ0(u)∪Γ+(u) = Γ. Note that ΓW = Γ0(u) (except for a set of measure zero) and Γ_F ⊂ Γ₋(u). In Figure 1.1 (from [Gri01], p.98) the situation of an opened window is sketched, which is described by (1.14) withσn=0.

Inflow and outflow is a consequence of thermal buoyancy effects. It is worth mentioning that in almost every application the so-called neutral zone, consisting of points located in the opening with u·n= 0, is of measure zero. A survey on boundary conditions for the

Y

Z X

domain of influx

neutral plane window opening

Figure 1.1.: Inflow at outflow regions at an opened window.

isothermal Navier-Stokes equations and further references thereon can be found in [Lia99].

More details on boundary conditions regarding the simulation of indoor-air movement can be found in [Nei99] and [Gri01].

The partition Γ₋(u), Γ0(u) and Γ+(u) is used for imposing boundary conditions for the temperature equation. It seems natural to require

θ=θin on Γ₋(u), a∇θ·n= 0 on Γ+(u) , whereθ_in designates the outside (fluid) temperature.

Depending on the physical boundary conditions at the wall, we consider the following sub-partitioning of Γ0(u), videlicet,

θ = θ_w on Γ_W,D , a∇θ·n = ˙qc⁻¹_p on Γ_W,N , (1.18)

whereθ_w denotes the wall temperature and ˙q denotes the heat-flux at the wall. Of course, Γ_W,D∩Γ_W,N =∅, Γ_W,D∪Γ_W,N = Γ_W .

1. The laminar model