Nusselt number

Heat transfer

To start, we define the heat transfer: surface-integrating the internal energy balance in the homogeneous directions yields

\[\sumzc \sumyc \frac{J}{\sfact{1}} \pder{T}{t} = \sumzc \sumyc \frac{1}{\sfact{1}} \dif{}{\gcs{1}} \left( - \frac{J}{\sfact{1}} \vel{1} \ave{T}{\gcs{1}} + \frac{1}{\sqrt{Pr} \sqrt{Ra}} \frac{J}{\sfact{1}} \frac{1}{\sfact{1}} \dif{T}{\gcs{1}} \right).\]

Assuming the flow field achieves a statistically-steady state

\[\pder{T}{t} \rightarrow 0\]

leads to

\[\frac{1}{\sfact{1}} \dif{}{\gcs{1}} \left\{ \sumzc \sumyc \left( \frac{J}{\sfact{1}} \vel{1} \ave{T}{\gcs{1}} - \frac{1}{\sqrt{Pr} \sqrt{Ra}} \frac{J}{\sfact{1}} \frac{1}{\sfact{1}} \dif{T}{\gcs{1}} \right) \right\} = 0,\]

where the wall-normal differentiation and the homogeneous summations are interchanged, which is justified for rectilinear coordinates. We introduce

\[\heattransfer \equiv \sumzc \sumyc \left( \frac{J}{\sfact{1}} \vel{1} \ave{T}{\gcs{1}} - \frac{1}{\sqrt{Pr} \sqrt{Ra}} \frac{J}{\sfact{1}} \frac{1}{\sfact{1}} \dif{T}{\gcs{1}} \right),\]

which is the internal energy going through a specific wall-normal position per unit time (heat transfer). Note that the sign is decided such that normal cases \(\vat{T}{\frac{1}{2}} > \vat{T}{\ngp{1} + \frac{1}{2}}\) give positive value.

The computation of the heat transfer on the walls \(\heattransfer\) are implemented as follows:

src/logging/heat_transfer.c

  const double diffusivity = fluid_compute_temperature_diffusivity(fluid);
  // on the bottom and top walls
  double energies[2] = {0., 0.};
  BEGIN
    const double hx_xm = HXXF(        1);
    const double hx_xp = HXXF(isize + 1);
    const double jd_xm = JDXF(        1);
    const double jd_xp = JDXF(isize + 1);
#if NDIMS == 2
    const double dt_xm = - T(    0, j) + T(        1, j);
    const double dt_xp = - T(isize, j) + T(isize + 1, j);
#else
    const double dt_xm = - T(    0, j, k) + T(        1, j, k);
    const double dt_xp = - T(isize, j, k) + T(isize + 1, j, k);
#endif
    energies[0] -= diffusivity * jd_xm / hx_xm / hx_xm * dt_xm;
    energies[1] -= diffusivity * jd_xp / hx_xp / hx_xp * dt_xp;
  END
  const size_t nitems = sizeof(energies) / sizeof(energies[0]);
  const void * sendbuf = root == myrank ? MPI_IN_PLACE : energies;
  void * recvbuf = energies;
  MPI_Reduce(sendbuf, recvbuf, nitems, MPI_DOUBLE, MPI_SUM, root, comm_cart);

By further integrating the differential equation in the wall-normal direction, we obtain

\[\sum_{i = 1}^{\chi} \sfact{1} \frac{1}{\sfact{1}} \dif{\heattransfer}{\gcs{1}} = - \vat{\heattransfer}{\frac{1}{2}} + \vat{\heattransfer}{\xi + \frac{1}{2}} = 0,\]

indicating that \(\heattransfer\) is constant for all wall-normal cell faces (\(\frac{1}{2}, \frac{3}{2}, \cdots, \ngp{1} - \frac{1}{2}, \ngp{1} + \frac{1}{2}\)).

Nusselt number

We focus on how much the heat transfer is enhanced compared to the reference case \(\heattransfer_{ref}\) if the flow fields were stationary with the given \(Ra\) and \(Pr\). For stationary flow fields, heat is purely conducted diffusively (i.e., \(\vel{1} \equiv 0\)), and the temperature profile is linear in the wall-normal direction:

\[\frac{1}{\sfact{1}} \dif{T}{\gcs{1}} = - 1 \,\, (\because \text{boundary conditions}),\]

and thus the reference heat transfer is given by

\[\heattransfer_{ref} = \sumzc \sumyc \frac{1}{\sqrt{Pr} \sqrt{Ra}} \frac{J}{\sfact{1}}.\]

The Nusselt number is defined as the ratio of them:

\[Nu \equiv \frac{\heattransfer}{\heattransfer_{ref}}.\]

As proved in the global balance of squared temperature, \(\heattransfer\) is linked to the source and sink of the squared temperature relation.

Squared Temperature

Now we revisit the relations derived in the global balance of squared temperature:

\[\dhinjall.\]

Due to the Dirichlet boundary condition with respect to the temperature, we can extract \(T\) out of summation symbols to obtain

\[\vat{ T }{\frac{1}{2}} \vat{ \heattransfer }{\frac{1}{2}} - \vat{ T }{\ngp{1} + \frac{1}{2}} \vat{ \heattransfer }{\ngp{1} + \frac{1}{2}}.\]

Additionally, since we fix the temperature difference of the two walls

\[\vat{T}{\frac{1}{2}} - \vat{T}{\ngp{1} + \frac{1}{2}} \equiv 1,\]

and \(\heattransfer\) is equal at every wall-normal cell faces

\[\heattransfer \equiv \vat{\heattransfer}{\frac{1}{2}} = \vat{\heattransfer}{\ngp{1} + \frac{1}{2}},\]

we notice

\[\heattransfer = \dhinjall,\]

and of course

\[\heattransfer = \dhdisall.\]

Squared Velocity

Integrating the definition of heat transfer in the wall-normal direction yields

\[\sumxc \sfact{1} \heattransfer = \sumzc \sumyc \sumxc J \vel{1} \ave{T}{\gcs{1}} - \sumzc \sumyc \sumxc \frac{1}{\sqrt{Pr} \sqrt{Ra}} \frac{J}{\sfact{1}} \dif{T}{\gcs{1}}.\]

The left-hand side is equal to \(\heattransfer\) since \(\sumxc \sfact{1} \equiv 1\) (recall that we assume the wall-normal length of the domain is unity). The second term in the right-hand side leads to

\[ \begin{align}\begin{aligned}& - \sumzc \sumyc \frac{1}{\sqrt{Pr} \sqrt{Ra}} \frac{J}{\sfact{1}} \sumxc \dif{T}{\gcs{1}}\\= & - \sumzc \sumyc \frac{1}{\sqrt{Pr} \sqrt{Ra}} \frac{J}{\sfact{1}} \left( \vat{T}{\ngp{1} + \frac{1}{2}} - \vat{T}{\frac{1}{2}} \right)\\= & \sumzc \sumyc \frac{1}{\sqrt{Pr} \sqrt{Ra}} \frac{J}{\sfact{1}}\\= & \heattransfer_{ref},\end{aligned}\end{align} \]

since \(J / \sfact{1}\) is independent to the homogeneous directions.

Thus we notice that

\[\heattransfer = \sumzc \sumyc \sumxc J \vel{1} \ave{T}{\gcs{1}} + \heattransfer_{ref},\]

which relates the Nusselt number with the squared velocity relations.