# Two-Dimensional Periodic Hartmann Flow with Electrically Insulating Walls
## 1. Problem description
This case models incompressible, electrically conducting flow in a two-dimensional channel. The streamwise boundaries are periodic, the top and bottom boundaries are stationary no-slip walls, and those walls are electrically insulating. A uniform transverse magnetic field couples the flow to an inductionless magnetohydrodynamic (MHD) electric-potential equation. A constant streamwise volumetric momentum source drives the flow. Table 1 summarizes the active model choices; every value in the table is taken from the supplied XML input.
**Table 1. Active physical and discretization models.**
| Setting | Configured value |
|---|---:|
| Element block | `eblock-0_0` |
| Physics block | `FluidPhysicsBlock` |
| Equation-set type | `IncompressibleNavierStokes` |
| Inductionless MHD equation | Enabled |
| Constant momentum source | Enabled |
| Temperature equation | Not enabled |
| Basis order | $1$ |
| Integration order | $2$ |
The XML hierarchy that activates the models in Table 1 is:
```xml
```
Table 2 lists the active physical parameters. Density is not specified in the XML, so the prescribed fallback $\rho=1.0\ \mathrm{kg\,m^{-3}}$ is assumed and clearly distinguished from extracted input values. The constant source is written as a volumetric momentum source because it enters the conservative momentum residual; its SI unit is therefore $\mathrm{N\,m^{-3}}$. The inactive constant-flow-rate control and its ancillary target/area entries are retained in the XML excerpt for completeness, but their units are not stated in the source and they are not used because the control is disabled.
**Table 2. Physical parameters used by the case.**
| Quantity | Symbol | Value |
|---|---:|---:|
| Kinematic viscosity | $\nu$ | $0.01\ \mathrm{m^2\,s^{-1}}$ |
| Fluid density (assumed fallback) | $\rho$ | $1.0\ \mathrm{kg\,m^{-3}}$ |
| Artificial-compressibility coefficient | $\beta$ | $100.0\ \mathrm{m^2\,s^{-2}}$ |
| Electrical conductivity | $\sigma$ | $1.0\ \mathrm{S\,m^{-1}}$ |
| External magnetic field | $\mathbf{B}^{0}$ | $(0.0,1.0,0.0)\ \mathrm{T}$ |
| Volumetric momentum source | $\mathbf{s}_{m}$ | $(4.194528049,0.0)\ \mathrm{N\,m^{-3}}$ |
The exact XML settings represented in Table 2 are:
The density row in Table 2 has no corresponding XML entry because it is the prescribed fallback used only when density is absent; the remaining rows are supported by the following exact settings.
```xml
```
```xml
```
### Nondimensional project conditions
The project uses the channel half-height $a=1.0\ \mathrm{m}$ as its single characteristic length. With the uniform initial streamwise speed $U_c=1.0\ \mathrm{m\,s^{-1}}$, the configured $\nu=0.01\ \mathrm{m^2\,s^{-1}}$, $B_0=1.0\ \mathrm{T}$, and $\sigma=1.0\ \mathrm{S\,m^{-1}}$, together with the prescribed fallback $\rho=1.0\ \mathrm{kg\,m^{-3}}$, the flow conditions are $\mathrm{Re}_a=100$ and $\mathrm{Ha}_a=10$. Equations (6) and (7) give the single definitions and numerical substitutions. A Prandtl number is not applicable because the temperature equation is disabled, as shown in Table 1.
## 2. Governing equations
The `IncompressibleTimeDerivative` closure and the configured artificial-compressibility coefficient specify the complete artificial-compressibility (AC) continuity equation. Equation (1) is the AC form used for this case; no EDAC pressure-diffusion closure is selected in the XML.
$$\frac{1}{\beta}\frac{\partial p}{\partial t}+\nabla\cdot(\rho\mathbf{u})=0.\tag{1}$$
For the constant-density model, Equation (1) may equivalently be divided by $\rho$ to give $(\rho\beta)^{-1}\partial p/\partial t+\nabla\cdot\mathbf{u}=0$.
The incompressible momentum balance is given by Equation (2). It contains the constant volumetric source and the Lorentz force, but no buoyancy or other body-force model is enabled.
$$\rho\left[\frac{\partial\mathbf{u}}{\partial t}+(\mathbf{u}\cdot\nabla)\mathbf{u}\right]=-\nabla p+\rho\nu\nabla^2\mathbf{u}+\mathbf{s}_{m}+\mathbf{J}\times\mathbf{B}^{0}.\tag{2}$$
The inductionless MHD closure uses Ohm's law, written as Equation (3), to obtain the electric-current density from the electric potential, fluid velocity, and imposed field.
$$\mathbf{J}=\sigma\left(-\nabla\varphi+\mathbf{u}\times\mathbf{B}^{0}\right).\tag{3}$$
Charge conservation, $\nabla\cdot\mathbf{J}=0$, converts Equation (3) into the electric-potential equation shown as Equation (4).
$$\nabla\cdot\left(\sigma\nabla\varphi\right)=\nabla\cdot\left[\sigma\left(\mathbf{u}\times\mathbf{B}^{0}\right)\right].\tag{4}$$
The coupling back to Equation (2) is the Lorentz-force density in Equation (5).
$$\mathbf{f}_{L}=\mathbf{J}\times\mathbf{B}^{0}.\tag{5}$$
### Flow conditions
For this plane channel, the characteristic speed is the uniform initial streamwise speed, $U_c=1.0\ \mathrm{m\,s^{-1}}$, and the sole characteristic length is the channel half-height, $a=1.0\ \mathrm{m}$. Equation (6) gives the Reynolds number $\mathrm{Re}_a=100$.
$$\mathrm{Re}_a=\frac{U_ca}{\nu}=\frac{(1.0\ \mathrm{m\,s^{-1}})(1.0\ \mathrm{m})}{0.01\ \mathrm{m^2\,s^{-1}}}=100.\tag{6}$$
Equation (7) defines the Hartmann number using the same half-height $a$. With the configured magnetic-field magnitude $B_0=\lVert\mathbf{B}^{0}\rVert=1.0\ \mathrm{T}$, conductivity $\sigma=1.0\ \mathrm{S\,m^{-1}}$, viscosity $\nu=0.01\ \mathrm{m^2\,s^{-1}}$, and prescribed fallback density $\rho=1.0\ \mathrm{kg\,m^{-3}}$, the Hartmann number is $\mathrm{Ha}_a=10$.
$$\mathrm{Ha}_a=B_0a\sqrt{\frac{\sigma}{\rho\nu}}=(1.0\ \mathrm{T})(1.0\ \mathrm{m})\sqrt{\frac{1.0\ \mathrm{S\,m^{-1}}}{(1.0\ \mathrm{kg\,m^{-3}})(0.01\ \mathrm{m^2\,s^{-1}})}}=10.\tag{7}$$
Table 3 defines every symbol appearing in Equations (1)--(7), including the differential operators. The value of $\rho$ is the documented fallback assumption because the XML does not specify density.
**Table 3. Symbols used in Equations (1)--(7).**
| Symbol | Definition | SI unit |
|---|---|---:|
| $t$ | Time | $\mathrm{s}$ |
| $p$ | Lagrange-pressure field | $\mathrm{Pa}$ |
| $\beta$ | Artificial-compressibility coefficient | $\mathrm{m^2\,s^{-2}}$ |
| $\mathbf{u}$ | Velocity vector | $\mathrm{m\,s^{-1}}$ |
| $\rho$ | Fluid density; assumed as $1.0\ \mathrm{kg\,m^{-3}}$ because no input value is supplied | $\mathrm{kg\,m^{-3}}$ |
| $\nu$ | Kinematic viscosity | $\mathrm{m^2\,s^{-1}}$ |
| $\mathbf{s}_{m}$ | Constant volumetric momentum source | $\mathrm{N\,m^{-3}}$ |
| $\mathbf{J}$ | Electric-current density | $\mathrm{A\,m^{-2}}$ |
| $\mathbf{B}^{0}$ | Imposed external magnetic field | $\mathrm{T}$ |
| $\sigma$ | Electrical conductivity | $\mathrm{S\,m^{-1}}$ |
| $\varphi$ | Electric potential | $\mathrm{V}$ |
| $\mathbf{f}_{L}$ | Lorentz-force density | $\mathrm{N\,m^{-3}}$ |
| $U_c$ | Characteristic speed, taken as the initial streamwise speed | $\mathrm{m\,s^{-1}}$ |
| $a$ | Channel half-height | $\mathrm{m}$ |
| $B_0$ | Magnitude of the imposed magnetic field, $\lVert\mathbf{B}^{0}\rVert$ | $\mathrm{T}$ |
| $\mathrm{Re}_a$ | Reynolds number based on $a$ | Dimensionless |
| $\mathrm{Ha}_a$ | Hartmann number based on $a$ | Dimensionless |
The exact closure-model selection supporting Equations (1)--(5) is:
```xml
```
## 3. Geometry and mesh description
The XML defines an inline rectangular domain $0.0\ \mathrm{m}\leq x\leq5.0\ \mathrm{m}$ and $-1.0\ \mathrm{m}\leq y\leq1.0\ \mathrm{m}$. It is divided into one block containing $8\times32=256$ four-node quadrilateral (`Quad4`) elements. Thus the uniform element dimensions are $\Delta x=0.625\ \mathrm{m}$ and $\Delta y=0.0625\ \mathrm{m}$. Table 4 records the mesh values, and Figure 1 visualizes the resulting structured mesh and boundary types.
**Table 4. Inline geometry and mesh.**
| Quantity | Value |
|---|---:|
| Streamwise interval | $x\in[0.0,5.0]\ \mathrm{m}$ |
| Wall-normal interval | $y\in[-1.0,1.0]\ \mathrm{m}$ |
| Domain length | $5.0\ \mathrm{m}$ |
| Domain height | $2.0\ \mathrm{m}$ |
| Element type | `Quad4` |
| Number of elements in $x$ | $8$ |
| Number of elements in $y$ | $32$ |
| Total number of elements | $256$ |
| Number of mesh blocks in each direction | $1\times1$ |
Figure 1 is generated from the inline mesh definition below. The left and right sides are paired with the `y-all` periodic condition using a geometric matching tolerance of $10^{-8}\ \mathrm{m}$. No Exodus mesh file was supplied, so Exodus-specific node numbering, side-set IDs, and connectivity beyond the inline structured definition cannot be independently checked.
```xml
```
## 4. Initial and boundary conditions
The fields are initialized uniformly over `eblock-0_0`. Table 5 shows the initial values: zero Lagrange pressure, unit streamwise velocity, zero wall-normal velocity, and a constant electric potential.
**Table 5. Uniform initial conditions.**
| Field | Initial value |
|---|---:|
| Lagrange pressure, $p(\mathbf{x},0)$ | $0.0\ \mathrm{Pa}$ |
| Streamwise velocity, $u_x(\mathbf{x},0)$ | $1.0\ \mathrm{m\,s^{-1}}$ |
| Wall-normal velocity, $u_y(\mathbf{x},0)$ | $0.0\ \mathrm{m\,s^{-1}}$ |
| Electric potential, $\varphi(\mathbf{x},0)$ | $2.0\ \mathrm{V}$ |
In Table 5, $\mathbf{x}=(x,y)$ is the position vector with SI unit $\mathrm{m}$. The exact initial-condition syntax is:
```xml
```
At $y=1.0\ \mathrm{m}$ (`top`) and $y=-1.0\ \mathrm{m}$ (`bottom`), the `No-Slip` condition sets the stationary-wall velocity to $\mathbf{u}=\mathbf{0}\ \mathrm{m\,s^{-1}}$. `ElectricCurrentDensityInsulating` imposes zero normal current, summarized by Equation (8), where $\mathbf{n}$ is the outward unit normal and is dimensionless.
$$\mathbf{J}\cdot\mathbf{n}=0\ \mathrm{A\,m^{-2}}.\tag{8}$$
The left and right sides are periodic, as specified in the mesh excerpt in Section 3. The exact wall-condition syntax supporting Equation (8) is:
```xml
```
## 5. Solver configuration and convergence criteria
The spatial discretization uses first-order basis functions with second-order integration, as shown in the physics-block excerpt in Section 1. Time integration uses Backward Euler and a constant step-control mode. The configured interval is $0.0\ \mathrm{s}$ to $10000.0\ \mathrm{s}$ with a maximum of $100$ time indices. The initial and minimum time steps are both $0.0001\ \mathrm{s}$, while the maximum is $100.0\ \mathrm{s}$. Table 6 consolidates the nonlinear and linear solver choices and convergence limits.
**Table 6. Time-integration and solver configuration.**
| Category | Setting | Value |
|---|---|---:|
| Time integration | Stepper | Backward Euler |
| Time integration | Initial/final time | $0.0/10000.0\ \mathrm{s}$ |
| Time integration | Final time index | $100$ |
| Time integration | Minimum/initial/maximum step | $0.0001/0.0001/100.0\ \mathrm{s}$ |
| Time integration | Step-control mode | Constant |
| Time integration | Maximum absolute/relative error | $10^{-8}/10^{-8}$ |
| Nonlinear solver | Method | Newton, line-search based, full step |
| Nonlinear solver | Residual test | $\lVert\mathbf{F}\rVert_2\leq10^{-11}$ |
| Nonlinear solver | Maximum Newton iterations | $10$ |
| Nonlinear solver | Test combination | OR |
| Linear solver | Solver | Belos |
| Linear solver | Preconditioner | Ifpack2 Schwarz, overlap level $1$ |
| Subdomain solve | Solver | Amesos2 Tacho LU |
| Execution | Workset size | $256$ |
| Pseudo-time control | Initial/target CFL number | $1.0/100.0$ |
| Pseudo-time control | CFL transition | $20$ |
Here $\mathbf{F}$ is the assembled nonlinear residual vector; the XML does not define a physical SI unit for its combined norm, so the configured norm threshold in Table 6 is reported as a solver scalar. The absolute and relative error thresholds are likewise reported exactly as configured rather than assigned invented physical units. The CFL numbers, polynomial orders, iteration counts, overlap level, workset size, and time index are dimensionless.
The linear-solver and preconditioner settings in Table 6 come from:
```xml
```
The time-control settings in Table 6 are:
```xml
```
The Newton convergence criteria completing that hierarchy are:
```xml
```
The local-step and CFL-related controls referred to in Table 6 are selected by:
```xml
```
```xml
```
## 6. Numerical results
Numerical results of the Hartmann problem obtained using Vertex-CFD are presented in Figures 2 and 3. The computed velocity profile is nearly uniform in the central region of the domain, while sharp velocity gradients develop near the walls, commonly referred to as Hartmann walls. As the Hartmann number, $Ha$, increases, the core flow becomes progressively flatter and the near-wall boundary layers become thinner. In particular, the characteristic boundary-layer thickness, $\delta$, is inversely proportional to the Hartmann number, such that $\delta \propto Ha^{-1}$.
The Hartmann problem admits an analytical solution that can be derived from the momentum equation under the assumption of a steady, fully developed flow. This solution provides a suitable benchmark for assessing the accuracy of the numerical method. Accordingly, error norms are computed by comparing the numerical and analytical velocity profiles at different mesh resolutions. The corresponding convergence orders are also reported to evaluate the rate at which the numerical solution approaches the analytical solution under mesh refinement.
**Figure 2.** Velocity surface plots for $Re = 100$ and $H_a = 100$ with linear and quadratic basis functions.
**Figure 3.** Velocity line plots for $Re = 100$ and $H_a = 100$ with linear and quadratic basis functions.
The analytical solution for the velocity is a function of the Hartmann number
$H_a$:
```{math}
:label: eq-hartmann-exact-velocity
\mathbf{u}_{\mathrm{exact}}(\mathbf{r},t)
=
\left(
u_0
\frac{
\cosh(H_a)-\cosh\left(\dfrac{H_a y}{L}\right)
}{
\cosh(H_a)-1
},
0
\right),
```
where
```{math}
H_a = LB_0\sqrt{\frac{\sigma}{\rho\nu}},
\qquad
y\in[0,L].
```
The flow is driven by a pressure drop imposed through a volumetric source
term. The linear pressure drop $\frac{dp}{dx}$ is a function of the Hartmann
number $H_a$, the Reynolds number $Re$, and the surface-averaged velocity $U$:
```{math}
:label: eq-hartmann-pressure-drop
\frac{dp}{dx}
=
\lambda
\frac{H_a}{\tanh(H_a)}
\frac{U^2}{h},
```
where
```{math}
:label: eq-hartmann-friction-factor
\lambda
=
\frac{2H_a^2}{Re}
\frac{1}{
\dfrac{H_a}{\tanh(H_a)}-1
}.
```
For a Reynolds number of $100$, the linear pressure drop is computed, and the
resulting values are shown in {numref}`tab-hartmann-dpdx`.
```{table} Linear pressure drop for the Hartmann problem.
:name: tab-hartmann-dpdx
:align: center
| Case | $Re$ | $H_a$ | $\dfrac{dp}{dx}$ |
| ---: | ---: | ---: | ---: |
| 1 | 100 | 100 | 101.010101 |
```
An $h$-refinement and $p$-refinement study was performed for the two cases in
{numref}`tab-hartmann-dpdx` using a channel uniformly meshed with
quadrilateral and triangular elements. The $L_1$ error norm between the
numerical surface-averaged velocity and its exact solution,
$U=1.0\ \mathrm{m/s}$, was computed.
The results and corresponding orders of convergence are reported in
{numref}`tab-hartmann-p1-re100-ha100`,
{numref}`tab-hartmann-p2-re100-ha100`.Second-order accuracy is achieved when linear basis functions are used. With quadratic basis functions, the observed convergence order exceeds three, indicating superconvergent behavior. Such behavior may occur for problems characterized by strong localized gradients, such as those present within the thin Hartmann boundary layers.
```{table} Order of convergence for the Hartmann problem with $Re=100$, $H_a=100$, and linear basis functions ($p=1$).
:name: tab-hartmann-p1-re100-ha100
:align: center
| $N_y$ | $L_1$ error norm (quads) | Order (quads) | $L_1$ error norm (tris) | Order (tris) |
| ---: | ---: | ---: | ---: | ---: |
| 128 | 0.000535586 | — | 0.000534261 | — |
| 256 | 0.000132993 | 2.009767951 | 0.000132622 | 2.010220367 |
| 512 | $3.13744\times10^{-5}$ | 2.083692812 | $3.12732\times10^{-5}$ | 2.084323335 |
| 1024 | $7.45231\times10^{-6}$ | 2.073826005 | $7.42561\times10^{-6}$ | 2.074346137 |
```
```{table} Order of convergence for the Hartmann problem with $Re=100$, $H_a=100$, and quadratic basis functions ($p=2$).
:name: tab-hartmann-p2-re100-ha100
:align: center
| $N_y$ | $L_1$ error norm (quads) | Order (quads) | $L_1$ error norm (tris) | Order (tris) |
| ---: | ---: | ---: | ---: | ---: |
| 64 | 0.000226979 | — | 0.000209159 | — |
| 80 | 0.000104845 | 3.461329866 | $9.47078\times10^{-5}$ | 3.550615033 |
| 96 | $5.1625\times10^{-5}$ | 3.885878924 | $4.54439\times10^{-5}$ | 4.027599172 |
| 128 | $1.39733\times10^{-5}$ | 4.021815589 | $1.12814\times10^{-5}$ | 4.843266273 |
```