Hemodynamic Simulations Workbench:
An Open-Source JavaScript Framework for Pulmonary Artery Pressure Modeling

This workbench delivers an intuitive, browser-based environment for simulating pulmonary artery pressure (PAP) models, turning intricate hemodynamic equations into interactive visualizations. Built entirely in JavaScript and released as open source, it lets clinicians and researchers adjust model coefficients in real time and immediately see the impact on predicted PAP, thereby bridging the gap between theoretical hemodynamics and bedside decision-making. By prioritizing clarity, ease of use, and community extensibility, the platform embodies George Box’s reminder that “all models are wrong, but some are useful,” providing a practical tool that makes complex vascular dynamics both accessible and clinically meaningful.

BASE: nGeneHS.js

LBSM: nGeneHS_LBSM.js

Nozzle: nGeneHS_deLavalNozzle.js

Debugging Notes

Practical notes on the fully-working nGeneHS_LBSM.html (Written June 10, 2025)

A concise record is presented below to recall the minor HTML-only tweaks that remove console warnings, keep the visible interface unchanged, and guarantee automatic start-up of the LBSM simulation.

Main achievements

• Console remains clean. Hidden elements satisfy the base-class expectations, eliminating slider, button, and null listener warnings.
• Interface stays intact. Auxiliary elements are wrapped in display:none, so the user-facing layout is preserved.
• Auto-run works reliably. The background PNG now loads first, then the simulation is redrawn; colour traces stay visible without an extra click.

Hidden helpers added

The base class looks for two “Test Slope” sliders and one demo button. Silent placeholders were therefore inserted:

• #scaleTestSlope1_slider and #scaleTestSlope2_slider plus matching <output>s.
• #testButton_method1 (unused but expected).

All reside in a single wrapper block set to display:none.

Console messages explained

Sequence of events on page load

1. The DOM finishes parsing.
2. An nGeneHS_LBSM instance is created.
3. The PNG background loads and triggers a load event.
4. refreshSimulation() clears the canvas, redraws the grid, paints the PNG, and re-plots the yellow and blue traces above the image.

Outcome

No warnings appear, the simulation starts automatically, traces remain visible, and manual re-runs continue to function through the existing “Run Simulation” button.

Written on June 10, 2025

Hemodynamic Simulation Source Code

de Laval Nozzle Model

Hemodynamic Modeling Based on de Laval Nozzle Theory (Written June 21, 2025)

The de Laval nozzle is a classic convergent-divergent flow passage known for accelerating compressible fluids (such as gases) to high velocities. It consists of three distinct sections: a converging inlet, a narrow throat, and a diverging outlet. In theory, as gas flows through a de Laval nozzle under steady conditions, it accelerates to sonic speed at the throat and can further accelerate to supersonic speeds in the diverging section if sufficient pressure drop is available. In cardiovascular hemodynamics, blood flow in certain regions—especially around heart valves—exhibits geometric and functional similarities to the convergent–throat–divergent pattern of a de Laval nozzle. This document examines the theoretical relevance of de Laval nozzle models to blood flow, focusing on the subvalvular (converging), valvular (throat), and supravalvular (diverging) sections of cardiovascular flow. The discussion covers three conceptual mappings of the nozzle model to hemodynamics (analogy, surrogate modeling, and direct integration), the quasi-steady assumptions used to adapt compressible flow concepts to pulsatile incompressible blood flow, and the limitations of applying such models given physiological realities.

Mapping the de Laval Nozzle Concept to Cardiovascular Flow

In cardiovascular fluid dynamics, the flow path from the heart through a valve and into the arteries can be conceptually mapped to a convergent–divergent nozzle configuration. Three levels of mapping can be considered, each progressively more concrete: (1) a qualitative analogy, (2) a surrogate modeling approach, and (3) a directly integrated structural model. These mappings envision the subvalvular region as a convergent inlet, the valvular orifice as a narrow throat, and the supravalvular region (e.g., the proximal aorta) as a divergent diffuser. In all cases, the goal is to leverage the well-understood physics of nozzle flow to gain insight into or simplify the description of blood flow dynamics through cardiac valves or arterial stenoses.

1. Analogy: The Cardiovascular Nozzle

   As an analogy, the de Laval nozzle serves as a descriptive tool to illustrate how blood accelerates and decelerates through varying cross-sectional areas in the cardiovascular system. Consider the left ventricular outflow tract and aortic valve: during systole, blood is ejected from the left ventricle through the outflow tract (subvalvular region) which generally tapers toward the aortic valve. This tapering is akin to a converging nozzle section, causing the blood velocity to increase as the flow area narrows. The aortic valve orifice itself represents the throat of the nozzle – the point of minimum cross-sectional area – where the blood velocity reaches its peak and the pressure is at its lowest. Beyond the valve, the blood enters the ascending aorta and aortic root, which often expand in diameter (supravalvular region) analogous to a diverging nozzle section. Here, the blood flow slows down as the area increases, and some of the static pressure is recovered.

   Through this analogy, key hemodynamic phenomena become easier to visualize. For instance, the significant pressure drop across a stenotic (narrowed) valve can be understood via the Venturi effect: as blood speeds up in the tight orifice, static pressure must decrease. Likewise, the concept of vena contracta – the narrowest actual flow stream just downstream of a valve – mirrors the throat effect in a nozzle where the flow jet contracts and reaches maximum velocity. Downstream of a normal valve or mild stenosis (with a gradually expanding aortic root), the diverging geometry allows partial pressure recovery, similar to how a diffuser recovers pressure in a nozzle flow. This simple geometric analogy helps clinicians and researchers qualitatively reason about flow acceleration, jet formation, and pressure changes in conditions like aortic stenosis or other valvular lesions.

   Analogy Highlight: Just as gas in a de Laval nozzle accelerates through a narrowed throat and then decelerates in the expansion, blood flow picks up speed as it passes through a constricted valve and slows down as the vessel widens downstream. The converging–throat–diverging structure is a useful mental model for understanding where velocities peak and pressures drop in cardiovascular flow.

2. Surrogate Modeling Tool: Simplified Nozzle-Based Calculations

   Beyond qualitative description, the de Laval nozzle concept can be used as a surrogate modeling tool in theoretical hemodynamics. In this context, surrogate modeling means using the nozzle’s simplified physics and equations to approximate or represent the behavior of blood flow without directly simulating the full complexities of cardiovascular fluid dynamics. For example, engineers and physiologists might treat a heart valve or arterial stenosis as an “equivalent nozzle” to estimate flow rates and pressure drops. By applying conservation of mass and energy as one would in a nozzle, one can derive relationships such as the continuity equation and Bernoulli’s principle to link flow velocity, cross-sectional area, and pressure difference across a constriction.

   In practice, this surrogate approach underlies many diagnostic and modeling techniques. A prime example is the use of the Bernoulli equation to estimate the pressure gradient across a stenotic valve from measured blood velocity. In echocardiography, the maximal jet velocity \( v_{\text{max}} \) through a narrowed valve is used to approximate the pressure drop \( \Delta p \) via the simplified relation \( \Delta p \approx \tfrac12 \rho\, v_{\text{max}}^{2} \) (or, for \( \rho \approx 1 \, \mathrm{g\,cm}^{-3} \), the clinical shorthand \( \Delta p \approx 4 v^{2} \)). This formula is essentially a Venturi-nozzle calculation assuming inviscid, quasi-steady flow. Similarly, when determining the effective orifice area of a valve, clinicians apply the continuity equation (an incompressible-flow analog to mass conservation in a nozzle): \( A_{\text{valve}} = \dfrac{Q}{v_{\text{valve}}} \), where \( Q \) is the flow rate and \( v_{\text{valve}} \) is the blood velocity at the valve. Such computations treat the valve region as if it were a nozzle throat connecting two larger reservoirs (the ventricle and the aorta) and yield surprisingly accurate results for many purposes.

   Researchers also incorporate nozzle-like elements in analytical models or lumped-parameter simulations of the circulation. For instance, an arterial stenosis might be modeled as a short converging–diverging segment in a one-dimensional flow model, using equations from nozzle theory to capture the drop in pressure and change in velocity. This surrogate modeling is valuable because it simplifies complex 3-D fluid dynamics into tractable equations while still capturing the first-order effects of area change on flow behavior. However, it typically requires calibration or empirical correction (such as discharge coefficients or pressure-recovery factors) to account for differences between the ideal nozzle theory and real blood flow (e.g., energy losses due to viscosity and turbulence).

3. Direct Integration as a Structural Model Component

   The third conceptual mapping involves directly integrating the de Laval nozzle structure into the theoretical framework of cardiovascular modeling. Rather than using the nozzle concept only as an analogy or a separate calculation tool, this approach builds a model in which the geometry and equations of a convergent–throat–divergent nozzle are embedded as part of the cardiovascular system’s structure. In other words, the section of the circulation containing a valve or stenosis is explicitly modeled as a tapered tube with a narrow throat, and the flow equations are applied accordingly along that segment.

   For example, consider a one-dimensional (1D) model of blood flow in the aorta that includes the aortic valve. Using a de Laval nozzle basis, one can represent the left ventricular outflow tract, valve, and proximal aorta as a continuous channel with varying cross-sectional area \( A(x) \): decreasing from the ventricle toward the valve and increasing beyond it. By applying the Euler or Navier–Stokes equations in 1D along this channel, one directly leverages nozzle flow dynamics to compute how pressure and velocity evolve from the subvalvular region, through the valvular throat, and into the supravalvular region. In the limit of steady, inviscid flow with constant density, these equations reduce to familiar nozzle relationships (mass-flow conservation \( A\,v = \text{const} \) and energy conservation akin to Bernoulli’s law). Even when extended to unsteady or viscous conditions, the nozzle-shaped structural model provides a foundation to incorporate additional effects (like unsteady inertial terms or viscous losses) in a spatially distributed manner.

   Direct integration of the nozzle model is particularly useful for investigating theoretical scenarios or boundary conditions that are otherwise difficult to capture with simple lumped models. For instance, one can examine how a given valve area and aortic geometry together influence the peak achievable flow rate or the pressure pulse waveform by solving the flow governing equations in this structured geometry. This approach bridges the gap between a purely conceptual analogy and a full computational fluid dynamics simulation: it uses the structure of the nozzle to simplify the domain while retaining the essential spatial variability of area that drives the hemodynamics. It should be noted, however, that building such a model still demands careful attention to the realism of assumptions (e.g., whether to treat the fluid as incompressible, how to include wall flexibility, etc.), and often hybridizes the ideal nozzle theory with empirical corrections to remain physiologically accurate.

Quasi-Steady Assumptions for Pulsatile Flow Adaptation

Adapting compressible nozzle models to cardiovascular flow requires acknowledging two major differences: blood is effectively incompressible (density does not change appreciably with pressure), and blood flow is pulsatile (unsteady) rather than strictly steady. The concept of quasi-steady flow is often invoked to justify using steady-state relationships from compressible flow in the transient, incompressible context of the cardiovascular system. A quasi-steady assumption means that at each instant of time, the flow behaves as if it were in a steady state corresponding to the instantaneous conditions, neglecting the effects of fluid acceleration or history. This assumption is reasonable if the flow’s time-varying nature is slow enough that the fluid has time to adjust to near-steady conditions within each phase of the cycle.

In practice, the quasi-steady approach in hemodynamics allows one to apply formulas like the Bernoulli equation or continuity equation instantaneously during a heartbeat. For example, when blood accelerates through a valve during systole, a quasi-steady model assumes that the relationship between pressure and velocity at that moment is given by the steady Bernoulli principle \( \Delta p = \tfrac12 \rho\, v^{2} \) for the instantaneous velocity even though the velocity is changing over time. The justification lies in the timescales: if the cardiac cycle frequency is moderate and the Womersley number (a dimensionless number quantifying unsteady flow effects) is not too high, inertial and unsteady boundary-layer effects may be small compared to the dominant convective acceleration and pressure drop. Under those conditions, the flow at each time behaves nearly like a snapshot of a steady nozzle flow.

Using quasi-steady assumptions, one effectively borrows the compressible-flow nozzle framework by treating blood as a nearly incompressible fluid (Mach number \( M = v/c \approx 0 \)) and by treating each phase of the pulsatile cycle independently. The compressibility aspect is removed by assuming constant density, which simplifies the nozzle equations to their incompressible form. The time-dependent aspect is handled by updating the flow parameters (e.g., valve area, instantaneous flow rate) as functions of time while still using steady relations at each time step. In essence, the compressible-flow equations are reduced to an incompressible, time-parametrized form. This approach has been quite successful for capturing peak pressure drops and flow rates – for instance, predicting the maximum jet velocity through a stenotic orifice using the instantaneous pressure difference from ventricle to aorta. It also underlies many clinical indices where dynamic measurements are interpreted through steady formulas.

However, it is important to recognize that quasi-steady adaptation is an approximation. When flow changes rapidly (such as at the very onset of systole or during valve closure), unsteady inertial effects and fluid accumulation can produce pressure deviations that the steady formulas do not capture. For example, an accelerating flow requires extra pressure (an inertial term) beyond the steady Bernoulli prediction to account for the kinetic-energy buildup. Likewise, flow separation or reattachment can lag behind instantaneous flow changes. Despite these nuances, quasi-steady models remain a cornerstone in theoretical hemodynamics because they greatly simplify analysis. They allow the use of well-known nozzle and Venturi relationships in a pulsatile setting, provided one remains mindful of their approximate nature and corrects for unsteady discrepancies when necessary (e.g., via an added unsteady term or empirical factors at high-acceleration phases).

Theoretical and Physiological Limitations

While the de Laval nozzle model provides useful insights, several assumptions inherent in the nozzle theory do not hold exactly in the cardiovascular system. Applying this model to blood flow therefore comes with important limitations:

These limitations highlight that while nozzle-based models can capture the core mechanism of how area variation influences flow, they oversimplify many aspects of cardiovascular hemodynamics. It is crucial to account for viscosity (e.g., by including resistance or loss coefficients), wall elasticity (by coupling the nozzle model with compliance elements), and unsteady effects (by including added mass or unsteady pressure terms) when trying to apply the de Laval nozzle concept quantitatively to blood flow. In many cases, hybrid models are used – for instance, combining Bernoulli’s equation with an empirical discharge coefficient to correct for energy losses, or using a nozzle model for the mean behavior and superimposing wave dynamics for pulse propagation.

To mitigate these differences when modeling blood flow using nozzle analogies, one can:

• Include viscous losses: Add resistance terms or discharge coefficients to represent energy dissipation due to viscosity and turbulence.
• Include compliance: Couple the model with elastic wall elements (e.g., Windkessel or spring-damper components) to simulate how vessel expansion buffers pressure changes.
• Account for unsteady effects: Incorporate unsteady inertial terms or use the unsteady form of Bernoulli’s equation to capture the additional pressure needed for accelerating blood and any phase lags in flow development.
• Validate geometry assumptions: Use empirical data or higher-fidelity simulations to adjust the model if the actual vessel geometry causes flow separations or asymmetries not predicted by an ideal nozzle shape.

Geometry and Function of a De Laval Nozzle

Figure: Diagram of a convergent–divergent (de Laval) nozzle, showing how flow velocity \(v\) increases from the converging inlet to the diverging outlet while pressure \(p\) and temperature \(T\) decrease along the length. The narrowest cross-section (the throat) is where the flow reaches sonic speed (Mach \(1\)). Upstream of the throat, in the converging section, the flow is subsonic and accelerates as the area decreases. Downstream of the throat, in the diverging section, the flow becomes supersonic and continues to accelerate as it expands into a larger area.

A de Laval nozzle is a tube that is pinched in the middle, consisting of a converging section followed by a diverging section. This hourglass-shaped geometry is carefully designed to accelerate a compressible fluid (gas) to high speeds, including supersonic velocities. In the convergent part of the nozzle, the cross-sectional area decreases toward the throat, causing the subsonic incoming gas to speed up (much like a Venturi tube). At the throat (the minimum area), the gas reaches sonic conditions (Mach \(M=1\)) under choked-flow conditions. Beyond this point, the diverging section expands the flow: the increasing area allows the gas to continue accelerating if it has reached Mach \(M=1\) at the throat, thereby transitioning into a supersonic flow regime (Mach \(M>1\)). The de Laval nozzle thus converts the fluid’s internal energy into directed kinetic energy. As the gas accelerates through the nozzle, its pressure and temperature drop significantly due to the isentropic expansion, while its velocity increases dramatically. This principle is exploited in applications such as rocket engines and turbines to produce high-velocity jets for thrust.

1. Converging Section (Subsonic Inflow)

   The converging section of the nozzle is the upstream part where the cross-sectional area decreases toward the throat. In this region, the flow is subsonic (Mach < 1). For subsonic gases, a smaller area causes the flow to accelerate (since the mass flow is constant, the fluid must speed up to get the same mass through a tighter space). This is analogous to squeezing an incompressible fluid through a pipe contraction – the fluid speeds up – but with compressible gas there is the added effect of density decreasing as well. Throughout the convergent section, velocity increases, Mach number rises (approaching 1.0 at the throat), static pressure drops, and static temperature drops. The drop in pressure from the inlet to the throat provides the driving force that accelerates the gas. This phenomenon is essentially a manifestation of the Venturi effect in compressible flow: as the channel narrows, kinetic energy increases at the expense of pressure energy.

   Near the inlet of the convergent section, the flow Mach number is low, so compressibility is minor and the pressure is relatively high. As the gas moves through the narrowing passage, it accelerates and its Mach number may increase from, say, \(M \approx 0.2\) or \(0.3\) up toward \(M \approx 1\) at the throat. Because \(M<1\), the relation
   \( \frac{dA}{A} = (M^2 - 1)\frac{dv}{v} \)
   tells us \(dA<0\) gives \(dv>0\), confirming the acceleration. The pressure drop in the convergent nozzle can be estimated from Bernoulli-like considerations (with compressibility corrections via the isentropic relations). By the time the gas reaches the throat, it has been compressed and accelerated enough that its Mach number is nearly 1 and the flow is choked if sonic conditions are achieved. In summary, the converging section acts to pre-accelerate the flow subsonically, preparing it to reach Mach 1 at the throat.

   It is worth noting that if the pressure ratio across the nozzle (upstream total pressure vs. downstream pressure) is not large enough, the flow may remain entirely subsonic throughout and never choke. In that case, the nozzle simply behaves like a subsonic Venturi tube – flow accelerates in the converging part and then decelerates in the diverging part (with pressure recovering). Only when the flow actually reaches Mach 1 at the throat does the diverging section serve its supersonic acceleration role. In practical nozzle operation (such as rocket engines), the upstream pressure is sufficiently high compared to the exit pressure that the throat will choke. At choke, the throat pressure is fixed at the critical value (\( \approx 0.528,p_0 \) for air), and a further decrease in exit pressure cannot increase the mass flow through the throat – it will only allow expansion to supersonic speeds in the diverging section.

2. Nozzle Throat and Choked Flow

   The throat is the minimum-area point of the nozzle and serves as the pivotal location for the flow behavior. Under design conditions, the throat is where the gas velocity reaches sonic speed \( M = 1 \). This is the condition of choked flow, meaning the mass flow has hit its maximum for the given upstream stagnation conditions. At the throat, the flow’s Mach number is exactly 1, and the local pressure, temperature, and density are the critical values \( p^{*}, T^{*}, \rho^{*} \) as given by the isentropic formulas above. The occurrence of choked flow effectively decouples the flow upstream of the throat from the conditions downstream – information (pressure disturbances) cannot travel upstream through a sonic point. In other words, once choked, the mass flow is controlled solely by upstream conditions (total pressure, temperature, throat area) and is insensitive to further drops in downstream pressure.

   The physics at the throat can be understood by considering that as the flow accelerates toward Mach 1, the pressure gradient needed to continue accelerating the gas becomes very steep. To achieve exactly \( M = 1 \) at the throat, a certain minimum pressure ratio is required (the back pressure must be low enough). When this is met, the throat “locks in” sonic conditions. At Mach 1, the flow speed equals the local speed of sound, so the gas cannot respond to small downstream pressure fluctuations – the sonic point acts like a one-way barrier for pressure signals. This is why beyond a certain point, further lowering the pressure after the nozzle does not increase the flow rate; it only influences what happens in the diverging section.

   In summary, the throat is where the critical flow occurs. Engineers often design the throat size \( A_* \) to set the desired mass-flow rate (since \( \dot{m} \) at choke is proportional to \( A_* \) and \( p_0 /\sqrt{T_0} \) as shown earlier). The flow properties at the throat are also useful reference points: for instance, the Mach 1 condition is used to start the expansion fan for supersonic-flow calculations in the diverging part. It is also important to ensure the nozzle is operated at or above the critical pressure ratio for choked flow; otherwise the nozzle won’t produce supersonic exit velocities. If the throat is not choked (i.e., \( M \lt 1 \) there), the nozzle behaves just like an orifice or Venturi with subsonic flow throughout, and the advantages of the de Laval design (producing supersonic jets) are not realized.

3. Diverging Section (Supersonic Expansion)

   Downstream of the throat lies the diverging section, which opens up gradually. When the flow is choked at the throat \( M = 1 \) there, the gas entering the diverging section is at Mach just above 1 and supersonic expansion can occur. In the diverging section, the area \( A \) increases with distance, and since the flow is now supersonic \( (M \gt 1) \), the relation \(\dfrac{dA}{A} = (M^{2} - 1)\,\dfrac{dv}{v}\) tells us \( dA \gt 0 \) yields \( dv \gt 0 \): the supersonic flow accelerates further as the area widens. This is a counter-intuitive flip from subsonic behaviour – a bigger cross-section pushes a supersonic fluid to go faster. Physically, what happens is that the high-pressure, high-temperature gas that was choked at the throat now undergoes a rapid expansion into the larger volume of the diverging nozzle. This expansion to lower pressure causes a strong conversion of thermal energy into kinetic energy: the gas cools and its pressure drops while its velocity increases to supersonic levels.

   Throughout the diverging section, Mach number keeps rising above 1. For example, it might go from \( M = 1 \) at the throat to \( M = 2, 3 \), or higher by the nozzle exit, depending on how much the area expands and how low the exit pressure is. Correspondingly, the static pressure \( p \) falls continuously along the expansion, and static temperature \( T \) also falls. The flow speed can reach extremely high values (for instance, in rocket nozzles, exhaust velocities on the order of 2–3 km/s are common, corresponding to Mach numbers of 5 or more, depending on gas properties). The pressure ratio between the nozzle inlet (stagnation pressure) and exit is the driving factor for how far the expansion goes; a larger pressure drop (pushing towards a vacuum at exit) allows more expansion and higher exit Mach.

   If the nozzle is correctly expanded, the exit pressure will equal the ambient pressure. In many practical cases, especially for rockets at high altitude, the expansion may be under-expanded or over-expanded. Under-expanded means the gas could expand more (exit pressure is still above ambient), so the flow will continue to fan out outside the nozzle. Over-expanded means the expansion went too far (exit pressure is below ambient); in this case, a supersonic jet will experience compression shocks outside (or even inside) the nozzle to adjust the pressure upward to ambient. In extreme over-expansion, flow separation can occur inside the diverging section, causing a shock structure and potentially unsteady behaviour. Designers avoid severe over-expansion that might cause the flow to detach from the walls. Ideally, at the design condition, the diverging section expands the gas just enough that exit pressure matches ambient, yielding maximal thrust with a uniform supersonic flow at exit.

   In summary, the diverging section of a de Laval nozzle transforms the sonic flow from the throat into a high-Mach supersonic jet by allowing it to expand against a falling pressure. The combination of the convergent section (to accelerate subsonically to \( M = 1 \)) and the divergent section (to accelerate supersonically beyond \( M = 1 \)) is what enables a de Laval nozzle to efficiently produce extremely fast flows. Thermodynamically, most of the enthalpy drop (and thus temperature drop) of the gas occurs in the diverging section, as this is where the bulk of internal energy is converted to kinetic energy. By the nozzle exit, the gas pressure is much lower than at the inlet, and the gas has cooled significantly (often to a fraction of its stagnation temperature), having expended its energy to achieve a high-velocity stream.

Isentropic Flow Governing Equations

• Assumptions and One-Dimensional Flow

  To analyze flow through a de Laval nozzle, we typically make the following ideal assumptions for simplicity:

  • Steady, one-dimensional flow: The flow is in steady state (does not change with time) and is confined to essentially one dimension along the nozzle’s axis (flow properties are uniform across any cross-section).
  • Ideal gas, isentropic process: The gas is treated as ideal, and the expansion through the nozzle is isentropic (no heat transfer and no friction or dissipative losses). Thus, entropy remains constant and the process is reversible and adiabatic.
  • No external forces aside from pressure: The flow is horizontal along the nozzle axis, so gravity or other body forces are negligible; pressure forces and inertia dominate.
  • Compressible flow: The gas velocity can reach high values (typically Mach $>0.3$), so density changes with pressure are significant and must be accounted for (unlike in incompressible flow).

  Under these conditions, the flow can be modeled by the classical equations of compressible fluid dynamics, which relate how area variations influence velocity, density, pressure, and Mach number in the nozzle.

• Continuity and Mass Flow Rate

  Because the flow is steady and one-dimensional, the mass flow rate \( \dot m \) remains constant at every cross-section of the nozzle. The continuity equation expresses this as:

  $$ \dot m \;=\; \rho\,A\,v \;=\; \text{constant} $$

  where \( \rho \) is the local gas density, \( A \) is the cross-sectional area, and \( v \) is the flow velocity at that cross-section. This means that as the area \( A \) changes along the nozzle, the product \( \rho A v \) must remain constant. In sub-sonic flow a reduction in \( A \) forces an increase in \( v \); unlike an incompressible liquid, however, a gas also changes its density \( \rho \) as it accelerates. The mass flow can be rewritten in terms of Mach number \( M \) and the speed of sound \( a \) as \( \dot m = \rho A M a \). For a perfect gas, \( a = \sqrt{\gamma R T} \) (with \( \gamma \) the specific-heat ratio and \( R \) the gas constant).

  Choked flow. There is a maximum mass flow rate that the nozzle can pass, achieved when the flow at the throat becomes sonic (\( M = 1 \)). Once choked, further lowering the downstream pressure cannot increase \( \dot m \). Combining continuity with the isentropic relations for an ideal gas gives the mass flux at the throat:

  $$ \frac{\dot m}{A^{\*}} \;=\; \rho^{\*}\,a^{\*} \;=\; p_{0}\, \sqrt{\frac{\gamma}{R\,T_{0}}}\, \left(\frac{2}{\gamma+1}\right)^{\frac{\gamma+1}{2(\gamma-1)}}, $$

  where \( A^{\*} \) is the throat area and \( p_{0}, T_{0} \) are the stagnation pressure and temperature upstream. Quantities marked with a superscript \( ^{\*} \) are evaluated at sonic (critical) conditions. This expression shows that \( \dot m \) reaches its ceiling once the throat is choked.

• Area–Mach Number Relationship

  One of the most important results for variable-area flow is the relationship between cross-sectional area and the Mach number of the flow. By differentiating the continuity equation and using the compressible-flow relations, one can derive a differential relation between area change and velocity (or Mach-number change):

  $$ \frac{dA}{A} \;=\; (M^{2}-1)\,\frac{dv}{v}. $$

  • If \(M<1\) (subsonic flow), then \(M^{2}-1\) is negative, so \(dA/A\) and \(dv/v\) have opposite signs. This means a reduction in area (\(dA<0\)) causes \(dv>0\), i.e. velocity increases – a converging duct accelerates subsonic flow. Conversely, expanding the area slows down subsonic flow.
  • If \(M>1\) (supersonic flow), \(M^{2}-1\) is positive, so \(dA/A\) and \(dv/v\) have the same sign. An increase in area (\(dA>0\)) now causes \(dv>0\), i.e. a diverging duct accelerates supersonic flow. A converging section would slow down a supersonic flow (which is how supersonic wind tunnels use a diverging diffuser to decelerate flow).
  • If \(M=1\), the equation gives \(dA=0\) for any finite \(dv\); in fact, at exactly Mach 1, an infinitesimal area change would require an infinite change in velocity – this is the choke condition at the throat.

  This behavior explains the convergent–divergent nozzle design: subsonic gas accelerates in the convergent part (to reach \(M=1\) at the throat), and then the supersonic gas accelerates further in the divergent part. The precise quantitative relation between area and Mach number for an isentropic flow can be derived by combining the continuity and energy equations. The result is an expression for the area ratio \(A/A^{\*}\) as a function of Mach \(M\) (where \(A^{\*}\) is the area when \(M=1\)):

  $$ \frac{A}{A^{\*}} \;=\; \frac{1}{M}\, \left[ \frac{2}{\gamma+1}\, \Bigl(1+\tfrac{\gamma-1}{2}\,M^{2}\Bigr) \right]^{\frac{\gamma+1}{2(\gamma-1)}}. $$

  This area–Mach-number relation comes from the isentropic-flow equations. It indicates that for a given area ratio \(A/A^{\*}\), there are two possible Mach solutions: one subsonic (\(M<1\)) and one supersonic (\(M>1\)). At the throat, \(A/A^{\*}=1\) and indeed \(M=1\). For \(A > A^{\*}\), the flow could be either subsonic or supersonic, but the nozzle’s upstream and downstream pressure conditions select the physically realized branch (subsonic acceleration in the first part, then the supersonic branch beyond the throat). Notably, as \(A/A^{\*}\) becomes very large in the divergent section, \(M\) on the supersonic branch increases toward a high value (the flow can approach very high Mach numbers if the pressure ratio allows). On the sub-sonic branch, as \(A\) increases (beyond the throat in a diffuser), \(M\) would decrease toward low values. In practice, the convergent–divergent nozzle operates by following the subsonic branch up to \(M=1\) at the throat, then transitioning to the supersonic branch in the expansion.

• Pressure, Density, and Temperature Relations

  The isentropic flow assumption provides important relations between pressure, density, temperature, and Mach number along the nozzle. As the gas accelerates and expands, its static pressure \(p\) and static temperature \(T\) drop, converting internal energy into kinetic energy, while the total (stagnation) values \(p_{0}\) and \(T_{0}\) remain constant (in an adiabatic reversible process). The following relations tie the local Mach number \(M\) to the ratios of static properties to their stagnation (reservoir) values:

  $$ \frac{p}{p_{0}} \;=\; \Bigl(1+\tfrac{\gamma-1}{2}\,M^{2}\Bigr)^{-\tfrac{\gamma}{\gamma-1}}, \qquad \frac{\rho}{\rho_{0}} \;=\; \Bigl(1+\tfrac{\gamma-1}{2}\,M^{2}\Bigr)^{-\tfrac{1}{\gamma-1}}, \qquad \frac{T}{T_{0}} \;=\; \Bigl(1+\tfrac{\gamma-1}{2}\,M^{2}\Bigr)^{-1}. $$

  These equations show, for example, that as \(M\) increases (due to acceleration in the nozzle), the pressure ratio \(p/p_{0}\) decreases and the temperature ratio \(T/T_{0}\) also decreases. At the throat (\(M=1\)) they reduce to the critical (sonic) ratios:

  • \(p^{\*} = p_{0}\,\Bigl(\tfrac{2}{\gamma+1}\Bigr)^{\tfrac{\gamma}{\gamma-1}}\),
  • \(T^{\*} = T_{0}\,\Bigl(\tfrac{2}{\gamma+1}\Bigr)\),
  • \(\rho^{\*} = \rho_{0}\,\Bigl(\tfrac{2}{\gamma+1}\Bigr)^{\tfrac{1}{\gamma-1}}\),

  (An asterisk denotes sonic conditions.) Thus, the pressure at the throat is a fixed fraction of the upstream total pressure (e.g.\ for \(\gamma = 1.4\), \(p^{\*} \approx 0.528\,p_{0}\)), and similarly for temperature and density. Throughout the convergent section the flow remains sub-sonic (\(M < 1\)), so pressure falls modestly and Mach rises toward 1. Once the flow becomes supersonic in the diverging section, \(M\) grows beyond 1 and the static pressure and temperature drop more sharply with further expansion. Because the flow is isentropic, the total values \(p_{0}\) and \(T_{0}\) stay constant from inlet to exit; all of the drop in static pressure and temperature is converted into kinetic energy.

Hemodynamic Analogy in Cardiovascular Flows

The principles of convergent–divergent nozzle flow have interesting parallels in cardiovascular hemodynamics, especially when considering blood flow through regions of varying cross-section such as heart valves and arterial segments. In the circulatory system, blood is an incompressible fluid, and vessel geometry can create nozzle-like effects. Notably, the flow through a stenotic heart valve (or any narrowed orifice in the vasculature) is often compared to flow through a nozzle: the blood accelerates through the constriction and then decelerates downstream. We can draw an analogy between the nozzle sections and specific anatomical regions:

Subvalvular region (ventricular outflow tract or vessel upstream of a stenosis): This is analogous to the converging section of a nozzle. As blood approaches a narrow valve opening or arterial stenosis, it is funneled from a wider chamber or vessel into a smaller orifice. The cross-sectional area available for flow decreases, so the blood velocity increases as it nears the valve. For example, in aortic stenosis, blood in the left ventricular outflow tract (LVOT) might have a velocity on the order of \(1~\text{m/s}\) or less, but as it is squeezed into the small valve orifice, it accelerates markedly. The flow is still subsonic in the hemodynamic sense (blood is effectively incompressible and \(M\approx 0\) with respect to the speed of sound in tissue), but the acceleration is analogous to a gas accelerating in a convergent nozzle. Accompanying this velocity increase, there is a pressure drop from the ventricle to just before the valve (much like the pressure drop in a convergent nozzle) – this is observed clinically as the proximal pressure in the ventricle being higher than the pressure in the outflow tract or just at the valve entrance when flow is occurring.

Valvular orifice (the valve opening or stenotic throat): This is analogous to the nozzle throat. The valve opening represents the point of minimum effective flow area – often referred to as the effective orifice area (EOA) in valvular stenosis. Here, the blood achieves its maximum velocity as it passes through the tightest constriction. In a severe aortic stenosis, for instance, the jet velocity through the valve can reach \(4~\text{m/s}\) or more. The flow is characterized by a large pressure gradient across the valve: upstream (ventricular) pressure is much higher than downstream (aortic) pressure during ejection, which corresponds to the conversion of pressure into kinetic energy in the jet. While there is no literal “Mach 1” condition (blood flow never comes anywhere close to the speed of sound in liquid, which is ~1500 m/s), one can think of an analogous “critical flow” concept in that the flow through the orifice is limited by the orifice area and driving pressure. If the valve area is very small, it acts as a limiting nozzle – the heart can only push so much cardiac output through for a given pressure. In fluid dynamics terms, the valve orifice causes a vena contracta effect: the blood jet actually contracts slightly just downstream of the physical orifice and reaches a peak velocity at the vena contracta (this is akin to the nozzle throat where velocity peaks). The pressure right at the vena contracta is lowest. Clinically, the maximum pressure drop (ΔP_max) occurs between the LVOT (just before the valve) and the vena contracta just beyond the valve.

Supravalvular region (the downstream artery or vessel): This corresponds to the diverging section in the analogy. Beyond the valve or stenosis, the blood flows into a larger cross-sectional area (the ascending aorta or pulmonary artery, or simply a wider part of the vessel after a stenosis). Consequently, the blood decelerates and some of its kinetic energy is converted back into pressure – this is the phenomenon of pressure recovery. In the artery just downstream, the cross-sectional area is greater than the valve opening, so the high-velocity jet expands, slows down, and the static pressure in the fluid partially rises again. This is analogous to a subsonic diffuser (since blood flow remains well below sonic speeds): after a constriction, the flow spreads out and pressure increases. However, unlike an ideal diffuser, there are losses due to turbulence in the blood jet mixing with the stagnant blood in the aorta, so not all pressure is recovered. Still, the net effect is that the pressure drop from the ventricle to the distal aorta (often called ΔP_recovery) is less than the initial drop at the valve itself (ΔP_max), because some pressure is regained in the expansion region. This is similar to how an ideal divergent nozzle would recover pressure if the flow remained subsonic, except in our case the flow through the valve is like a free jet expanding into the aorta rather than a confined diffuser. The compliance of the artery also plays a role: as the high-speed blood enters, the artery can elastically expand to accommodate the surge of volume (particularly in the case of the aorta which has significant elasticity). This expansion (due to compliance) effectively increases the cross-sectional area available and aids in reducing the peak pressure and smoothing the flow – an effect with no direct analog in a rigid nozzle.

Hemodynamic Flow Equations and Analogous Variables

From a mathematical perspective, blood flow through a constricted region can be described by equations analogous to those used for nozzle flow, with appropriate modifications for the incompressible and pulsatile nature of blood. Some key correspondences and differences are outlined below:

In the above comparison, we see that many qualitative behaviors align: a narrowing causes acceleration and pressure drop, and a subsequent expansion allows deceleration and partial pressure rise. The major distinctions arise from compressibility versus compliance. In gases, the ability to compress (change density) is what allows the flow to reach sonic speeds and beyond, and to have a choke point. In blood (which is nearly incompressible), the analogous “give” in the system is the elasticity of the vessels. The pulse wave speed in an elastic artery (which is the speed at which pressure disturbances travel) acts like the sound speed in a gas. This wave speed \( c \) is given by Moens–Korteweg equation, \( c = \sqrt{\frac{Eh}{2\rho R}} \) for a vessel (where \( E \) is elastic modulus, \( h \) wall thickness, \( R \) radius), or in terms of area compliance \( C_A = \frac{dA}{dP} \), one can show \( c^2 \approx \frac{A}{\rho, dA/dP} \). In the human arteries, \( c \) is on the order of 5–10 m/s (much faster than blood flow), so the “Mach number” \( M_h = v/c \) stays very low (0.1–0.2 under peak flows). This means blood flow is subcritical and information travels upstream easily, preventing any choking phenomenon under normal conditions. Only in extreme situations (such as severe flow limitation in collapsible airways or perhaps shock waves in traumatic aortic ruptures) would a Mach-1-like condition occur in biological flows – these are rare and involve different mechanics (e.g., in airflow during a cough, the concept of a “choke point” does emerge).

To conclude, modeling the subvalvular, valvular, and supravalvular regions after a de Laval nozzle provides a useful conceptual and mathematical framework. The subvalvular outflow tract is like a convergent nozzle section accelerating the blood toward the valve. The valve or stenotic orifice is like a throat that creates a high-velocity jet (with an associated pressure drop). The supravalvular region (the receiving artery) behaves like a diffuser where the blood expands, slows down, and recovers some pressure. Equations analogous to those of nozzle flow – continuity (flow conservation) and energy (Bernoulli) – can be applied: for instance, using the continuity equation \( A_1 v_1 = A_2 v_2 \) to relate velocities in the ventricle and through the valve, or Bernoulli’s equation to relate pressure differences to velocity changes. Vessel compliance enters these equations by allowing the effective area to vary with pressure and by smoothing the pulsatile flow. Ultimately, while blood does not achieve supersonic speeds, the nozzle analogy helps in understanding phenomena like the formation of a high-speed jet through a stenosis, the drop and partial recovery of pressure, and the factors (area, pressure, compliance) that govern flow through cardiac valves and arterial narrowings.

Written on June 21, 2025

Integrating de Laval nozzle theory into lumped-branch hemodynamic simulation (Written June 22, 2025)

1 · Scope and motivation

Precise representation of pulsatile trans-valvular flow remains a central challenge in physics-based cardiovascular modelling. Classic lumped-branch systemic models (LBSM) partition the circulation into resistive (R), inertial (L) and compliant (C) elements; yet the rapid, area-dependent acceleration that dominates near the cardiac valves is handled only approximately. By contrast, a de Laval converging–diverging nozzle describes compressible flow through a variable-area conduit with analytically tractable momentum, energy and continuity relations. The present note examines how nozzle concepts can clarify, surrogate or formally extend LBSM valve segments while respecting physiological constraints.

2 · Re-examining the classical de Laval formulation

1. Canonical area–Mach relation

   For quasi-one-dimensional, isentropic flow of a calorically perfect gas, the throat-normalised area \(A/A^{*}\) and local Mach number \(M\) satisfy :contentReference[oaicite:0]{index=0}

   \[ \frac{A}{A^{*}}=\frac{1}{M}\Biggl[\frac{2}{\gamma+1}\Bigl(1+\frac{\gamma-1}{2}M^{2}\Bigr)\Biggr]^{\tfrac{\gamma+1}{2(\gamma-1)}} . \]

   Because the expression is implicit in \(M\), root-finding (e.g. Newton–Raphson) is required during simulation.

2. Pressure and temperature fields

   The isentropic static–to–total relations follow trivially: \[ \frac{T}{T_{0}}=\Bigl(1+\tfrac{\gamma-1}{2}M^{2}\Bigr)^{-1},\qquad \frac{p}{p_{0}}=\Bigl(1+\tfrac{\gamma-1}{2}M^{2}\Bigr)^{-\tfrac{\gamma}{\gamma-1}}. \] Choked flow occurs when \(M=1\) at the minimum area; further reductions in back pressure merely extend the supersonic portion downstream :contentReference[oaicite:1]{index=1}.

3. Issues detected in the current JavaScript prototype

   • Incorrect explicit Mach formula. The implemented square-root form neglects the implicit dependence in the canonical relation and omits the \(1/M\) factor.
   • Throat area mis-assignment. Astar is inconsistently referenced as AStar, and the special-case multiplier *100 at \(x=0\) alters the geometry.
   • Non-physical scaling factors. Multipliers of \(0.001\) (Mach) and \(1/2\) (velocity) distort dimensional coherence.
   • Coordinate shift. Translating \(x\) by \((x-0.3)\times10\) obscures the physical length scale and complicates coupling to LBSM time-bases.
   • Unused update hooks. updateComputedValues() is empty, preventing parameter changes from propagating to derived fields.

3 · Essential code corrections (excerpt)

Mach(x) {
  const A = this.A_geometry(x);
  const f = M => (A/this.Astar) -
               (1/M) *
               Math.pow((2/(this.Gamma+1))*(1+(this.Gamma-1)*M*M/2),
                        (this.Gamma+1)/(2*(this.Gamma-1)));
  return newtonRaphson(f, 0.2, 20);   // sub-routine not shown
}

V_nozzle(x){
  const M = this.Mach(x);
  const T = this.T0 / (1 + (this.Gamma-1)*M*M/2);
  const a = Math.sqrt(this.Gamma*287*T); // local speed of sound
  return M * a;
}

4 · Mapping nozzle sections to valvular flow domains

5 · Levels of integration

1. Pedagogical analogy

   Visualising transient valve flow as a miniature, mildly compressible nozzle highlights the transition from potential to kinetic energy and clarifies the role of exit pressure in regurgitant states.

2. Surrogate framework

   Within an LBSM time-step, substitute the valve element by a nozzle whose throat area equals the instantaneous effective orifice area \(A_{v}(t)\). The governing set becomes \[ Q = A_{v}(t) \, v,\quad \Delta p = p_{\text{up}}-p_{\text{down}} = \rho\!\left(\frac{v^{2}}{2}\right)\! \Bigl(f_{\gamma}(M) + \xi_{\!v}\Bigr), \] where \(f_{\gamma}\) derives from the isentropic nozzle relations and \(\xi_{\!v}\) covers viscous losses.

3. Formal structural coupling

   A dedicated nozzle module may replace the lumped L–R pair of the valve compartment. State variables include throat Mach number and chamber total pressure; they evolve through explicit, semi-implicit, or operator-splitting techniques while the surrounding LBSM retains its ordinary differential formulation in pressure and volume.

6 · Practical coupling strategy

1. Synchronise discretisation. Assign the nozzle solver the same Δt as the cardiovascular ODE integrator to eliminate interpolation.
2. Map parameters.
   • \(A^{*}(t)\) ← measured valve area (e.g. from echocardiography).
   • \(\gamma\approx1.01\) for blood (quasi-incompressible yet finite sound speed).
   • Total pressure \(p_{0}\) ← upstream ventricular pressure.
3. Iterative interface. Use Picard iteration to reconcile nozzle exit static pressure with LBSM downstream arterial pressure within each cardiac time-step.

7 · Expected benefits and limitations

• Improved prediction of early-systolic jet acceleration and peak velocity without ad-hoc Bernoulli coefficients.
• Natural accommodation of pressure-wave reflection when coupled to compliant downstream segments.
• Caveats: blood compressibility is weak; therefore \(\gamma\) approaches unity and numerical stiffness increases. Careful non-dimensional scaling is advised.

8 · Concluding remarks

Replacing or augmenting the LBSM valve compartment with a rigorously corrected de Laval-based module offers a physically consistent route to capture rapid trans-valvular dynamics while maintaining computational efficiency. Continued validation against Doppler and catheter data is encouraged before clinical deployment.

Written on June 22, 2025

Mapping de Laval-nozzle theory to pulmonary arterial hemodynamics: toward an enhanced LBSM framework (Written June 22, 2025)

Abstract

The de Laval nozzle provides a classical reference for compressible, one-dimensional, quasi-steady flow with an internal geometric contraction–expansion. The lesion–valve–post-valve (subvalvular–valvular–supravalvular) arrangement of the pulmonary outflow tract resembles this topology. By recasting nozzle relations into the Lumped-Biophysical System Model (LBSM), improved interpretability of pressure–velocity waveforms may be obtained, particularly for pulmonary arterial pressure (PAP) estimation. The following note (i) corrects internal inconsistencies in the supplied JavaScript nozzle routine, (ii) establishes a disciplined variable mapping between nozzle and cardiovascular domains, and (iii) delineates the conceptual limits of the analogy given blood’s low-Mach-number, weakly compressible behaviour.

1 Refinement of the de Laval-nozzle implementation

1. Geometry

   The original A_geometry routine introduces variable-name mismatch (AStar vs. Astar) and a throat multiplier (×100) that breaks continuity. A consistent definition is recommended:

   \[ A(x)=A^{*}\times\begin{cases} 1-k\,x, & x<0,\\[4pt] 1+k'\,x^{2}, & x\ge 0, \end{cases} \qquad x\in[-L_{\text{conv}},\,L_{\text{div}}]. \]

   • All symbols maintain lower-case convention; \(A^{*}\) denotes the throat area.
   • Continuity of \(A\) and its derivative at \(x=0\) is preserved (C¹ continuity) if \(k=k'\,x|_{0^{-}}\), or a spline is used.

2. Mach-number evaluation

   Scaling factors 0.001 (Mach), /10 (temperature), and /2 (velocity) obscure physical meaning. Removal is suggested. Using the isentropic area–Mach relation:

   \[ \left(\frac{A}{A^{*}}\right)^{2}= \frac{1}{M^{2}} \left[ \frac{2}{\gamma+1} \left(1+\frac{\gamma-1}{2}M^{2}\right)\right]^{\frac{\gamma+1}{\gamma-1}}. \]

   A numerical root-finder (e.g. bisection or Newton–Raphson) avoids analytic inversion error. The pressure-ratio term \(\bigl(p_{e}/p_{0}\bigr)\) must be employed for the choking check, not introduced directly into every pointwise Mach evaluation.

3. Recommended code fragment

   Use a pure function machFromArea(areaRatio, γ) returning the physical root. Remove any unit-less scaling, and employ SI units consistently (m, kg, s).

2 Variable correspondence between nozzle flow and pulmonary outflow

In the LBSM, the pulmonary arterial velocity function \[ v_{\text{pul}}(t)=\alpha\,e^{-\gamma_{R} t}(1-t/ET)\,t \] already resembles the temporal profile obtained from integrating the nozzle momentum equation under quasi-steady conditions. Substituting \(t\mapsto x/c\) (convective travel time) establishes a direct mapping for forward-propagating waves.

3 Composite valvular model using nozzle sections

1. Sub-valvular (converging)

   • Area contraction \(A(x)\lt A_{0}\) raises static pressure and accelerates the flow.
   • Equivalent to the muscular infundibulum; stiffness modulation is represented by parameter \(k\).

2. Valvular throat

   • Geometry fixed by leaflet opening; chosen as \(A^{*}(t)\).
   • Choking condition analogue: maximum transmitral or trans-pulmonary gradient before regurgitation.

3. Supra-valvular (diverging)

   • Gradual expansion relaxes velocity; wave reflection occurs at branch points.
   • Parameter \(k'\) links to local compliance \(C_{\text{PA}}\).

4 Integration into LBSM

1. Replace empirical \(\alpha,\,\beta,\,\gamma\) coefficients with nozzle-derived expressions using \(\gamma,\,k,\,k'\).
2. Express \(A_{\text{eff}}(t)\) through measured valve orifice dynamics to capture patient-specific stenosis.
3. Insert the refined velocity into the pressure ODEs of LBSM, maintaining conservation of mass–momentum.
4. Validate against catheter-derived PAP and Doppler flow in a cohort with varying pulmonary vascular resistance.

5 Limitations of the nozzle analogy

• Compressibility: Blood behaves as a nearly incompressible Newtonian fluid at \(M\approx10^{-3}\). The nozzle model requires compressibility for energy conversion. This discrepancy mandates reinterpretation of Mach-related terms as dimensionless inertial indices rather than true Mach numbers.
• One-dimensionality: The nozzle neglects secondary flow and wall shear; pulmonary arteries exhibit complex three-dimensional pulsatility and viscoelastic wall behaviour.
• Rigid walls in nozzle theory: LBSM incorporates compliance via \(C\), but a pure nozzle does not. Coupling must therefore include a compliance term or employ fluid–structure interaction corrections.
• Turbulence onset: Transitional Reynolds numbers in large vessels (Re ≈ 2000) differ from gas-dynamic criteria; least-squares calibration against clinical data remains essential.

6 Concluding remarks

The nozzle framework, when prudently corrected and cautiously interpreted, affords an analytically tractable surrogate for rapid forward modelling of pulmonary flow and pressure. Embedding its geometry-dependent relations into the LBSM allows direct linkage between measurable valvular geometry and global haemodynamics, advancing patient-specific assessment of right-ventricular afterload.

Written on June 22, 2025

Integrating de Laval-nozzle gas dynamics into lumped-parameter blood-flow modeling (Written June 22, 2025)

1  Scientific context

Modern lumped-parameter hemodynamic simulators (e.g. LBSM) treat the cardio-vascular tree as a network of resistive (R), compliant (C) and inertial (L) elements, yet the rapid, high-Reynolds acceleration through valvular regions remains difficult to capture. Compressible-flow theory developed for the de Laval nozzle offers an analytical scaffold for these high-velocity segments. Although blood is essentially incompressible, the mathematical structure of the nozzle equations (area variation ⇄ velocity amplification ⇄ pressure adaptation) can be re-interpreted for liquid flow by substituting the Bernoulli–unsteady terms for the isentropic Mach relations.

2  Segment-by-segment correspondence

3  Re-examining the de Laval implementation

1. Geometry definition

   The JavaScript routine A_geometry(x) multiplies the throat area Astar by a linear or quadratic factor, yet two issues distort the physical picture:

   • At x = 0 a mistaken identifier (AStar) and an arbitrary factor 100 inflate the area.
   • All positions are rescaled via (x – 0.3)*10; this hides the true axial distance and can generate negative areas.

   A robust, non-dimensional formulation is recommended:

   \( A(x)= \begin{cases} A^*\!\left[1-k \dfrac{x}{L_c}\right], & -L_c\le x &lt; 0 \\ A^*, & x = 0 \\ A^*\!\left[1+k' \bigl(x/L_d\bigr)^2\right], & 0 &lt; x \le L_d \end{cases} \)

2. Area–Mach relation

   For isentropic flow the exact algebraic link is

   \( \dfrac{A}{A^*} = \dfrac{1}{M}\! \left[\dfrac{2}{\gamma+1}\! \left(1+\dfrac{\gamma-1}{2}M^2\right)\right]^{\frac{\gamma+1}{2(\gamma-1)}} \)

   No closed-form inversion exists; Newton–Raphson or pre-tabulation is therefore preferable. The present code’s simplification (A/A*)2/(γ-1) ⇄ M² under-predicts supersonic values and then scales them by 0.001, pushing the returned Mach number well below unity. For haemodynamics substitute \(M\) with a dimensionless velocity ratio \( \mu = V/\sqrt{\Delta P/\rho}\), where \( \Delta P\) is the instantaneous valve-upstream pressure head.

3. Velocity and pressure fields

   • Gas dynamics: \( V = M \sqrt{\gamma R T_0 (1+(\gamma-1) M^2/2)^{-1}} \)
   • Blood flow analogue: \( V = \sqrt{2(\,P_{\text{prox}}-P_{\text{dist}}\,)/\rho} \, f\!\left(\!\dfrac{A}{A^*}\!\right) \) with f captured from the tabulated area–ratio curve.

4  Embedding in LBSM

1. Time-invariant (steady) variant

   1. Replace the valvular R element with a nonlinear conductance \(G(A)=\rho^{-1}A^{-2}f(A/A^*)^{-2}\).
   2. Update the nodal equations once per cardiac-cycle using the mean valve area (echocardiographic planimetry).

2. Quasi-steady pulsatile variant

   1. Treat the cardiac period as a series of frozen-flow instants (e.g. 1–2 ms).
   2. For each instant fetch the instantaneous pressure drop \( \Delta P(t) \) from the LBSM state vector.
   3. Compute \( V(t) \) via the adapted nozzle look-up and feed it back to update flow \( Q(t)=A(t)V(t) \).
   4. Advance the ordinary-differential solver; the additional algebraic step costs < 5 % run-time.

5  Implementation checklist

• Drop the *0.001 and /2 scaling of Mach() and V_nozzle().
• Rename AStar → Astar at the throat and restore area = Astar.
• Remove the opaque coordinate transform; handle axial length in SI units and let the canvas routine apply its own pixels-per-metre conversion.
• Tabulate \(A/A^* \mapsto \mu\) once at start-up for \(\gamma=1.3\,\text{–}\,1.4\); cubic-spline interpolation suffices (128 points).
• Export the tabulated function so that the LBSM core can call it without referencing the canvas subsystem.

6  Sensitivity overview

7  Concluding remarks

Embedding nozzle-based acceleration into lumped-parameter hemodynamics reconciles two scales: microscopic valve jet physics and macroscopic cardio-vascular load. The corrected implementation eliminates non-physical scaling, restores geometric consistency, and provides an extensible bridge for both steady and quasi-steady simulations. Future work may refine the divergence function to accommodate turbulent dissipation or couple the jet momentum to compliant post-valvular segments.

Written on June 22, 2025