Boundary patches on a block

Boundary conditions in ember are applied through patches: labelled regions on the faces of a ember.block.Block that carry inlet, outlet, periodic or rotating-wall data. This example builds an annular duct block, attaches a patch to each face, configures the boundary data, and draws the block with its patches coloured by type.

Patches index a constant-\(i\), \(j\) or \(k\) face of the block, so the parent block must already have its coordinates set before a patch is attached – the patch needs the geometry to identify the span and pitch directions of the surface of revolution.

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.patches import Patch as LegendPatch

import ember.block
import ember.patch
from ember import util

Annular duct geometry

A block spanning a short axial length of an annulus: i runs axially, j radially across the span, and k circumferentially across one pitch. The coordinates are set first so that patches can attach to it.

block = ember.block.Block(shape=(8, 5, 7))
block.set_xrt(util.linmesh3((0.0, 1.0), (0.9, 1.1), (0.0, 0.35), block.shape))

<ember.block.Block object at 0x74ca70530bd0>

Creating and attaching patches

Each patch names the constant face it occupies (i=0 is the first axial plane, i=-1 the last) and an optional label. Omitting the other two indices spans the whole face. Patches live in the block’s patches collection and can be added one at a time or in bulk.

block.patches.append(ember.patch.InletPatch(i=0, label="inflow"))
block.patches.append(ember.patch.OutletPatch(i=-1, label="outflow"))
block.patches.extend(
    [
        ember.patch.PeriodicPatch(k=0, label="lower"),
        ember.patch.PeriodicPatch(k=-1, label="upper"),
    ]
)
block.patches.append(ember.patch.RotatingPatch(j=0, label="hub"))

# Patches are retrievable by index, by label, or by type.
print(f"{len(block.patches)} patches attached")
print("by label:", block.patches["inflow"])
print(
    "by type: ",
    len(block.patches.inlet),
    "inlet,",
    len(block.patches.periodic),
    "periodic",
)

# Once attached, the constant dimension, shape and point count are known.
for patch in block.patches:
    print(
        f"  {patch.label:>8}: {type(patch).__name__:14} "
        f"const_dim={patch.const_dim} shape={patch.shape} size={patch.size}"
    )

5 patches attached
by label: InletPatch(i=(0, 0), j=(0, -1), k=(0, -1), label='inflow')
by type:  1 inlet, 2 periodic
    inflow: InletPatch     const_dim=0 shape=(1, 5, 7) size=35
   outflow: OutletPatch    const_dim=0 shape=(1, 5, 7) size=35
     lower: PeriodicPatch  const_dim=2 shape=(8, 5, 1) size=40
     upper: PeriodicPatch  const_dim=2 shape=(8, 5, 1) size=40
       hub: RotatingPatch  const_dim=1 shape=(8, 1, 7) size=56

Configuring boundary data

Inlet patches take stagnation conditions and flow angles, uniform or varying across the patch (any input broadcasting to the patch shape). Outlet patches take a static pressure, optionally throttled to a target mass flow with a PID controller. Rotating patches take an angular velocity.

inlet = block.patches["inflow"]
inlet.set_Po_To_Alpha_Beta(Po=2e5, To=1200.0, Alpha=0.0, Beta=0.0)

# A radial stagnation-temperature profile across the span (j direction).
_, nj, _ = inlet.shape
To_profile = np.linspace(1100.0, 1300.0, nj).reshape(1, nj, 1)
inlet.set_Po_To_Alpha_Beta(To=To_profile)
print(f"Inlet To varies {float(inlet.To.min()):.0f}--{float(inlet.To.max()):.0f} K")

outlet = block.patches["outflow"]
outlet.set_P(1e5)  # Static pressure [Pa]; an initial guess if throttling
outlet.set_throttle(mdot_target=3.0, K_pid=(1.0, 0.1, 0.0))  # PID throttle to mdot

hub = block.patches["hub"]
hub.set_Omega(500.0)  # rad/s
print(f"Hub speed: {hub.rpm:.0f} rpm")

Inlet To varies 1100--1300 K
Hub speed: 4775 rpm

Extracting data on a patch

Every patch exposes a slice that indexes its face out of the parent block, either as a 2-D sub-block (with all properties available) or applied directly to a property array.

inlet_face = block[inlet.slice]  # 2-D sub-block on the inlet face
print(f"Inlet face block shape: {inlet_face.shape}")
print(
    f"Same coordinates direct from the array: "
    f"{np.array_equal(inlet_face.x, block.x[inlet.slice])}"
)

Inlet face block shape: (1, 5, 7)
Same coordinates direct from the array: True

Visualising the patched block

Drawing each patch on its face shows how the boundaries tile the block. A light wireframe of the whole block places each patch extent within the full domain – including the casing face, which carries no patch here. Cartesian coordinates follow from \(y = r\cos\theta\), \(z = r\sin\theta\); each patch face is one singleton-dimension slice.

colours = {
    "InletPatch": "C0",
    "OutletPatch": "C3",
    "PeriodicPatch": "C2",
    "RotatingPatch": "C1",
}


def to_xyz(b):
    """Cartesian coordinates of a block, with any singleton axes removed."""
    x, r, t = np.asarray(b.x), np.asarray(b.r), np.asarray(b.t)
    return x.squeeze(), (r * np.cos(t)).squeeze(), (r * np.sin(t)).squeeze()


def block_edges(b):
    """The twelve edges of a block, as sub-blocks tracing the actual nodes."""
    ends = (0, -1)
    edges = [b[:, j, k] for j in ends for k in ends]  # along i
    edges += [b[i, :, k] for i in ends for k in ends]  # along j
    edges += [b[i, j, :] for i in ends for j in ends]  # along k
    return edges


fig = plt.figure(figsize=(7, 5))
ax = fig.add_subplot(111, projection="3d")

# Wireframe of the whole block (curved circumferential edges traced), drawn
# heavier than the patch faces are translucent so the full extent reads through.
for edge in block_edges(block):
    ax.plot(*to_xyz(edge), color="0.3", lw=1.2)

for patch in block.patches:
    X, Y, Z = to_xyz(block[patch.slice])
    ax.plot_surface(
        X,
        Y,
        Z,
        color=colours[type(patch).__name__],
        alpha=0.55,
        edgecolor="k",
        linewidth=0.2,
    )

ax.set_xlabel("$x$ [m]")
ax.set_ylabel("$y$ [m]")
ax.set_zlabel("$z$ [m]")
ax.view_init(elev=20, azim=125)

legend = {type(p).__name__: p for p in block.patches}  # one entry per type
ax.legend(
    handles=[LegendPatch(facecolor=colours[name], label=name) for name in legend],
    loc="upper left",
)

fig.tight_layout()
plt.show()