'nefes'1 Transport of carried scalars on edges
The network settles pressure between components by negotiation — an element’s pressure relation couples the states on its two sides — but temperature is not settled that way. The gas in a duct is as hot as wherever it came from; total enthalpy is carried with the flow rather than balanced across a component. This document builds the one relation that expresses that fact, shows that it is the single member of a family that also carries species and any other advected scalar, and explains why mass alone is deliberately left out of the family.
The device has to meet the same robustness standard as the rest of the mean-flow solver (see state and recovery): it must stay well defined while the solver is still discovering the flow direction, iterate smoothly through a reversal, and reduce to exact upwinding once the flow has settled. Meeting that standard is what turns the plain statement “each edge inherits its upstream neighbour’s enthalpy” into the smoothly upwinded, donor-based form derived here.
1.1 Standing assumptions
The relations below are posed at a steady operating point, under the overview assumptions, of which two are load-bearing here.
- Section-averaged edge scalars. Each carried quantity takes a single representative value on an edge, the section average of the framework; the true port flux of a convected scalar \psi is A\,\langle m\,\psi\rangle = \beta_\psi\,\dot m\,\psi_e, and the transport relation is written for the edge value \psi_e with the profile-shape factor \beta_\psi absorbed into the closure.
- Adiabatic elements unless a source is declared. An element neither adds nor removes the carried scalar except through an explicit lumped source (a flame’s heat release, an inline injector), so that in the absence of such a source the scalar is conserved through the element.
Throughout, an edge e carries the signed mass flow \dot m_e along its reference arrow, and \sigma_{P,e} = \pm 1 orients that arrow at element P (+1 if P is the tail, -1 if the head), so that the mass leaving P through e is \dot m^{\text{out}}_{P,e} = \sigma_{P,e}\dot m_e and the mass entering is \dot m^{\text{in}}_{P,e} = -\sigma_{P,e}\dot m_e.
1.2 The donor enthalpy of an element
We first ask what total enthalpy an element makes available to the streams leaving it, a quantity we call its donor enthalpy H_P. Steady energy conservation at element P, in the flux form of the governing equations, is given as:
\sum_e \sigma_{P,e}\,\dot m_e\, h_{t,e} = 0,
where the sum runs over the edges incident to P and h_{t,e} is the total enthalpy carried on edge e. Splitting the sum into inflow ports, where h_{t,e} is set by the upstream neighbour, and outflow ports, where the gas leaves carrying the element’s donated value H_P, and using mass conservation to equate the two rate sums, yields the donor as the inflow-mass-weighted average of the incoming enthalpies:
H_P \;=\; \frac{\sum_e \dot m^{\text{in}}_{P,e}\,h_{t,e}}{\sum_e \dot m^{\text{in}}_{P,e}} \qquad\text{(sum over inflow ports, } \dot m^{\text{in}}_{P,e} > 0\text{)},
where \dot m^{\text{in}}_{P,e} = -\sigma_{P,e}\dot m_e is the mass entering P through edge e. It can be interpreted as the perfect-mixing statement of a lumped element: having no resolved interior, the element blends its incoming streams into one mixed state and offers that single state to every outlet — all outflow ports of one element therefore carry the same total enthalpy. How the mass then splits between several outlets is set by the pressure relations, not by energy; the thermodynamic state of the mixed stream is single-valued.
Worked example (a tee). Two streams mix and leave through one outlet, with inflows \dot m_1 = 3\,\mathrm{kg/s} at T_{t,1} = 300\,\mathrm{K} and \dot m_2 = 1\,\mathrm{kg/s} at T_{t,2} = 500\,\mathrm{K}, so that h_{t,i} = c_p T_{t,i}. The donor is given as:
H_P = \frac{3\,(c_p\cdot 300) + 1\,(c_p\cdot 500)}{3 + 1} = c_p\cdot 350\,\mathrm{K},
that is T_{t,\text{out}} = 350\,\mathrm{K}, pulled toward the larger stream as mixing demands; were there two outlets, both would leave at 350\,\mathrm{K}.
The exact-upwind average above is not yet solver-safe, because “inflow port” is a branch on the sign of \dot m_e and the denominator vanishes at zero throughflow. Both are removed by replacing the hard positive part with its smooth counterpart, the same C^\infty primitive family used throughout the residuals (see complex-step):
H_P \;=\; \frac{\sum_e w_e\,h_{t,e}}{\sum_e w_e}, \qquad w_e = \operatorname{spos}\!\big(\dot m^{\text{in}}_{P,e};\,\varepsilon\big) = \tfrac{1}{2}\Big(\dot m^{\text{in}}_{P,e} + \sqrt{(\dot m^{\text{in}}_{P,e})^2 + \varepsilon^2}\,\Big),
where w_e is the smooth donor weight of edge e, \operatorname{spos}(\cdot;\varepsilon) is the regularized positive part, and \varepsilon is the smoothing width in mass-flow units. Because w_e \ge \varepsilon/2 > 0, the mixture is well defined for any flow state, degrading gracefully to a plain average at complete stagnation, while at a converged state the weight of an outflow port decays like \varepsilon^2/(4\,|\dot m_e|): a quadratically small contamination.
Boundary elements override this donor with their specification rather than mixing incoming streams: an inlet offers its prescribed c_p T_t^{\text{spec}}, and a pressure outlet offers c_p T_t^{\text{backflow}}, the total enthalpy a re-entering stream would carry — the same convention adopted at pressure outlets in field solvers.
1.3 The edge transport equation
With each element offering a donor value, each edge contributes one equation that delivers the donor of whichever element is upstream onto that edge. Writing \operatorname{tail}(e) and \operatorname{head}(e) for the elements at the tail and head of the arrow, the edge transport equation is given as:
h_{t,e} \;=\; \theta(\dot m_e)\, H_{\operatorname{tail}(e)} \;+\; \big(1 - \theta(\dot m_e)\big)\, H_{\operatorname{head}(e)}, \qquad \theta(x) = \tfrac{1}{2}\Big(1 + \frac{x}{\sqrt{x^2 + \varepsilon^2}}\Big),
where \theta is a smoothed Heaviside step (the regularized upwind weight): for clearly positive \dot m_e it tends to 1 and the edge inherits the tail element’s donor, for clearly negative to 0 and it inherits the head’s, and near zero it blends the two — keeping the relation well-behaved at the one state where the physics genuinely cannot decide which side is upstream.
Three properties carry the whole design, and the first is the reason the transport lives on the edge rather than being reassigned between an element’s ports.
- No degeneracy. The equation always carries a coefficient of 1 on h_{t,e} itself, so it stays independent of the balances as the flow tends to zero and cannot collapse onto another row of the system (contrast the near-degeneracy that a port-assigned convected condition would develop at rest, discussed in well-posedness).
- Smoothness. The weight \theta has no kink anywhere, so the residual and its Jacobian are smooth and complex-analytic through a flow reversal, and the solver may iterate through \dot m_e = 0 without a special case.
- Asymptotic exactness. At a converged state with |\dot m_e| \gg \varepsilon the smooth weight differs from the exact upwind value by \mathcal{O}\!\big((\varepsilon/\dot m_e)^2\big): the leak onto the downstream side is 1 - \theta \approx \tfrac{1}{4}(\varepsilon/\dot m_e)^2, so with the default \varepsilon = 10^{-4}\,\dot m_{\text{ref}} the imprint on the solution sits near the 10^{-8} relative level, far below any engineering tolerance (test:
test_passive_tracer_mixes_mass_weighted, verifying the mass-weighted donor mix).
These first and third properties are visible directly in the shipped weight, plotted in Figure 1: the left panel shows how \theta rounds the hard upwind switch over a blend region of width set by \varepsilon, and the right panel confirms that the downstream leak decays with the advertised quadratic slope, so the converged solution is exact to the smoothing order.
Code
eps_show = [0.15, 0.05] # exaggerated, in units of mdot_ref, so the blend is visible
fig = make_subplots(
rows=1, cols=2,
subplot_titles=("smooth upwind weight θ(ṁ)", "downstream leak 1 − θ, ṁ > 0"),
)
# left: theta vs the hard Heaviside
x = np.linspace(-0.6, 0.6, 600)
fig.add_trace(go.Scatter(x=x, y=np.where(x > 0.0, 1.0, 0.0), mode="lines",
line=dict(color="#888888", width=1.5, dash="dot"),
name="hard upwind"), row=1, col=1)
for k, eps in enumerate(eps_show):
theta = np.array([float(smooth_step(xi, eps)) for xi in x])
fig.add_trace(go.Scatter(x=x, y=theta, mode="lines", line=dict(color=COLORWAY[k]),
name=f"ε = {eps:g} ṁ_ref"), row=1, col=1)
# right: the downstream leak and its quadratic reference
xp = np.logspace(-1.0, 1.0, 400) # ṁ / ε, so the curves overlay
for k, eps in enumerate(eps_show):
md = xp * eps
leak = np.array([1.0 - float(smooth_step(m, eps)) for m in md])
fig.add_trace(go.Scatter(x=md, y=leak, mode="lines", line=dict(color=COLORWAY[k]),
name=f"1 − θ, ε = {eps:g}", showlegend=False), row=1, col=2)
ref = 0.25 * (eps / md) ** 2
fig.add_trace(go.Scatter(x=md, y=ref, mode="lines",
line=dict(color=COLORWAY[k], width=1, dash="dash"),
name="¼(ε/ṁ)²" if k == 0 else None,
showlegend=(k == 0)), row=1, col=2)
fig.update_xaxes(title_text="ṁ / ṁ_ref", row=1, col=1)
fig.update_yaxes(title_text="θ", row=1, col=1)
fig.update_xaxes(title_text="ṁ", type="log", row=1, col=2)
fig.update_yaxes(title_text="1 − θ", type="log", row=1, col=2)
figsmooth_step kernel, in mass flow normalized by a reference \dot m_\mathrm{ref}. Left: \theta rounds the hard Heaviside upwind switch (grey) over a blend region whose width is set by the smoothing \varepsilon; two exaggerated values are shown so the blend is visible — the default \varepsilon = 10^{-4}\,\dot m_\mathrm{ref} would render as an indistinguishable step at this scale. Right: the leak onto the downstream side, 1 - \theta for \dot m_e > 0, on log axes; it follows the reference slope \tfrac{1}{4}(\varepsilon/\dot m_e)^2 (dashed), the quadratically small contamination that makes the converged transport exact to \mathcal{O}((\varepsilon/\dot m_e)^2).
Finally, no separate entropy transport is needed alongside the enthalpy one. For a single-component gas the entropy is fixed by the stagnation pair (p_t, T_t) (the entropy lemma of state and recovery), both of which are already carried — p_t through the pressure relations and T_t through this enthalpy transport — so entropy is a derived quantity rather than an independently advected one.
1.4 Generalization to any carried scalar
Nothing in the construction used a property of enthalpy beyond its being an intensive scalar passively advected with the flow: a quantity whose edge value is inherited from the upstream element and whose junction value is the mass-weighted mixture of the inflows. Any per-unit-mass scalar \phi — a species mass fraction, a conserved mixture fraction Z_i, a reaction-progress or burnt marker, a passive tracer — therefore obeys the identical pair, with a donor \Phi_P and an edge equation given as:
\Phi_P \;=\; \frac{\sum_e w_e\,\phi_e \;+\; \dot S^{\phi}_P}{\sum_e w_e}, \qquad \phi_e \;=\; \theta(\dot m_e)\,\Phi_{\operatorname{tail}(e)} \;+\; \big(1 - \theta(\dot m_e)\big)\,\Phi_{\operatorname{head}(e)},
where \phi_e is the scalar carried on edge e, \Phi_P the donor value element P offers, w_e and \theta are the same smooth weights as for enthalpy, and \dot S^{\phi}_P is any lumped source of \phi within the element. This is not an aspiration but the literal shape of the assembly: a single loop over the carried scalars applies the same donor-and-upwind kernel to the total enthalpy (\phi = h_t) and to each transported composition scalar in turn. Total enthalpy is simply the first instance, with \phi = h_t, \Phi_P = H_P, and \dot S^{\phi}_P = \dot Q_P the lumped heat-release rate; the mixture-fraction transport of reacting flow is the subject of thermochemistry.
Each new scalar adds exactly one unknown per edge and one transport equation per edge, so the squareness of the system is preserved automatically, without any change to the element rows (see equation structure).
Sources enter through the donor, not the edge. The source term \dot S^{\phi}_P appears only in the donor — the mixed value becomes “what flowed in, plus what the element added, spread over the throughflow mass” — while the per-edge equation is untouched. This is the same hook by which a boundary element overrides its donor: because the element owns its donor, sources, specifications, and overrides all live in one place and none of them alters the equation count. A heat-addition flame is the archetype, raising the outflow total enthalpy by \dot Q_P/|\dot m| through its enthalpy donor, its |\dot m| floored by the same \varepsilon so the jump stays bounded and smooth at zero flow (see elements). An important caveat is that a fixed source with vanishing net throughflow has no bounded steady donor: as \sum_e w_e falls to its regularization floor, \dot S^{\phi}_P / \sum_e w_e grows large, and while the floor keeps the value finite rather than undefined it becomes \varepsilon-dependent. That regime — steady heating of a stagnant junction — is genuinely ill-posed physics, not a solver artifact.
1.5 Why mass is not a carried scalar
Mass conservation deliberately stays a nodal balance and is not recast as an edge donor–transport pair, for two independent reasons.
- Nothing to carry. The donor–upwind device transports an intensive, per-unit-mass scalar whose edge value is inherited from upstream. The mass flux \dot m_e is the extensive flux itself — the carrier, not the carried — and it is the edge unknown; the analogue of \phi for mass is the constant 1, whose transport equation would be vacuous.
- No defect to repair. The per-edge form exists to remove the direction-dependent equation count of the convected conditions (see equation structure). The mass balance is already direction-invariant in count — one per element, with \sigma_{P,e}\dot m_e merely changing sign under an arrow flip — so there is no structural discontinuity to absorb; recasting it would only forfeit exact global conservation and reintroduce the very \dot m \to 0 degeneracy the edge form was built to avoid.
The clean statement of the framework is therefore that mass is conserved nodally, while every intensive carried scalar — total enthalpy today, species and progress variables in a reacting network — is transported on the edges by the donor-and-upwind pair of this document.
The pressure relations that give each element its physics are the subject of elements.