Estimation of pulsatile energy dissipation in intersecting pipe junctions using inflow pulsatility indices

This study aims to characterize the effect of inflow pulsatility on the hydrodynamic power loss inside intersecting double-inlet, double-outlet pipe intersection (DIPI) with cross-flow mixing. An extensive set of computational fluid dynamics (CFD) simulations was performed in order to identify the individual effects of flow pulsatility parameters, i.e., amplitude, frequency, and relative phase shift between the inflow waveform oscillations, on power loss. An experimentally validated second order accurate solver is employed in this study. To predict the pulsatile flow performance of any given arbitrary inflow waveforms, we proposed three easy-to-calculate pulsatility indices. The frequency-coupled quasi-steady flow theory is incorporated to identify the functional form of pulsatile power loss as a function of these indices. Our results indicated that the power loss within the inflow branch sections, lumped outflow-junction section, and the whole conduit correlates strongly with the pulsatility of each inflow waveform, the total inflow pulsatility, and inflow frequency content, respectively. The complete CFD simulation matrix provided a unified analytical expression that predicts pulsatile power loss inside a one-degree offset DIPI geometry. The predictive accuracy of this expression is evaluated in comparison to the CFD evaluation of arbitrary multi-harmonic inflow waveforms. These results have important implications on hydrodynamic pipe networks that employ complex junctions as well as in the patient-to-patient comparison of surgically created vascular connections. Coupling the present analytical pulsatile power loss expression with non-dimensional steady power loss formulation provided a valuable predictive tool to estimate the pulsatile energy dissipation for any arbitrary junction geometry with minimum use of the costly CFD computations.


I. INTRODUCTION
Cross-flow jet flows, where two jet wakes from opposed nozzles impinge, are widely used in industrial applications, such as polymer processing, 1 nanoparticle synthesis, 2 and stream reactors, [3][4][5][6][7] to enhance fluid mixing efficiency and combustion. Associated hydraulic energy dissipation has been studied extensively in relation to traditional junctions with simple topologies that are used as a core application of fluid dynamics in underground pipe networks, [8][9][10][11][12][13] liquid distribution lines, 14 and branching networks, 15 where the optimum system efficiency requires improved internal flow conditions and optimal hydrodynamic design at these pipe intersections. In a medical context, patients with univentricular heart defects require a series of palliative surgical operations, such as the Fontan procedure, which involves the use of opposed-jet vascular connections for total cavopulmonary connection (TCPC). [16][17][18][19][20][21][22][23][24] The local flow dynamics of this junction have been extensively studied in order to minimize energy dissipation within the connection, [25][26][27][28] which relieves the elevated post-operative systemic resistance and improves the efficiency of the single ventricle circuit. 29 Dynamic instabilities and oscillatory behavior within the junction zone have been well documented for the opposed-jet flows under the laminar inlet flow regime, 3,6,[30][31][32] and the introduction of an offset between the inflow branches has demonstrated higher stability 33 and lower energy dissipation. 25,[33][34][35][36] Steady-state energy dissipation inside patient-specific TCPC geometries has been successfully defined using non-dimensional metrics such as the steady inflow rate, boundary surface area, pulmonary flow split (FS), and anatomical features. [37][38][39] However, comparatively limited research attention has been given to the pulsatile flow characteristics within TCPC geometry, which closely relate to the pronounced respiratory effects 40,41 and compensated exercise response 16,42 of single ventricle circulation.
Our previous investigations of pulsatile inlet (caval) flow waveforms at a fixed steady flowrate for total cardiac output (CO) 33 indicated that TCPC power loss is significantly influenced by the phase-angle between the inlet flows. For a standard junction geometry that employs a one diameter offset (1DO) double inlet pipe junction (DIPI), where the inflows are positioned one diameter apart [ Fig. 1(a)], the power loss decreases consistently as caval waveforms shift incrementally from the in-phase inflow state. These research findings about the dependency of dissipation on the phase difference justify further investigation into fluid dynamics to identify and quantify the pulsatile characteristics of caval waveforms and their relationship to junction power loss. It is significant to note that there has not been a rationale to assess the relative pulsatile energy efficiency for two different inflow waveform sets having equal-mean flowrates and different pulsatility conditions. However, delineation of the key pulsatility parameters that can correlate with the conduit energetics is potentially valuable to enable patient-to-patient comparison and hydrodynamic design in both pipeline transmission and clinical applications. Previously, a number of pulsatility indices have been proposed, mostly as clinically meaningful parameters to assess peripheral vascular disease and atherosclerotic occlusions, [43][44][45] renovascular function, 46,47 fetal cardiovascular function, 48,49 and arterial growth. 50,51 However, these imaging-based clinical pulsatility metrics have not been extended to the investigation of complex cardiovascular systems for which pulsatility is derived from the interaction of multiple inflows. New clinical metrics based on the caval inflow pulsatility will likely further complement the steady TCPC energy dissipation formulation 37 and provide analytical expressions to predict pulsatile TCPC power loss without the need for computational fluid dynamics (CFD) simulation.
The objective of this study is to characterize the effect of inflow pulsatility on the power loss within intersecting pipes using formal physically derived indices. In order to focus solely on the inflow pulsatility in the absence of intrinsic flow instabilities, an idealized TCPC geometry with the limited cross-flow impingement of a one diameter offset double inlet pipe junction (1DO DIPI) model was selected. The initial phase of this study represented the key pulsatility parameters and their influence on the inflow, junction, and outflow regions of the geometry based on single harmonic inflow waveforms. As detailed in Sec. II B, a set of simulations, where each parameter was altered one-at-a-time, was performed to quantify the individual effect of each of the pulsatility parameters and their sensitivity on power loss. An analytical expression incorporating the quasi-steady flow assumption was derived to identify the functional form of the pulsatile TCPC power loss in terms of the pulsatility indices. Based on the complete simulation matrix results, a unified analytical expression that predicts the pulsatile power loss inside TCPC (i.e., the 1DO DIPI model) was derived. The predictive accuracy of this expression was further evaluated in comparison to the CFD simulation of the arbitrary multi-harmonic inflow waveforms.

A. Computational model and solver settings
Computational simulations were performed using the idealized 1DO DIPI model with a diameter of 1.3 cm in order to examine the isolated effect of inflow waveform pulsatility on power loss. Tetrahedral unstructured grids were generated using ANSYS workbench (ANSYS Inc., Canonsburg, PA). A mesh sensitivity analysis was carried out at five grid resolutions to ensure a mesh independent solution. The five grid resolutions considered were as follows: (1) coarse grid with 150 K elements, (2) medium grid with 1. (a) Discretization of the one diameter offset double inlet pipe junction (1DO DIPI) geometry into three regions: inflow branches 1 and 2, the lumped junction-outflow section (i.e., outflow 1 and 2) and the junction zone according to regional pulsatile characteristics. (b) Time averaged velocity and pressure contours demonstrate the Poiseuille-like laminar flow regime within the 1DO DIPI geometry with mild stagnation and recirculation in the vicinity of the junction zone.

ARTICLE
scitation.org/journal/adv 250 K elements, (3) fine grid with 500 K elements, (4) finer grid with 750 K elements, and (5) finest grid with 1.1 M elements. The medium grid resolution was chosen for the computational simulations due to its close convergence of estimated power loss to the independent grid solution with optimum computational cost (see Sec. V of the supplementary material for detailed information on the grid sensitivity analysis). The governing equations for incompressible and Newtonian blood flow (ρ = 1060 kg/m 3 , μ = 3.71 × 10 −3 Pa s) were solved using a laminar unsteady solver based on finite-volume discretization in ANSYS (ANSYS, Inc., Canonsburg, PA). The continuity and linearized momentum equations were sequentially solved using an ANSYS segregated solver. A second-order up-winding scheme was utilized to discretize the convention terms in momentum equations due to the complexity of the flow regimes and to eliminate the numerical diffusion introduced by the discretization. The pressure-velocity coupling was considered to be PISO (Pressure Implicit with Splitting of Operators) for a better convergence rate considering the transient nature of the problem and the possibility of a high degree of mesh skewness. Due to the existence of rotational flow structures and curved geometry, the pressure interpolation was performed using the Pressure Staggering Option (PRESTO) method to acquire the pressure values at cell faces. Finally, the temporal integration of the solution was accomplished using a second-order implicit scheme. These solver settings have been demonstrated to provide flow-field solutions in good quantitative agreement with particle image velocimetry experiments, both for time-averaged flow fields and transient unsteady flow features in different complex TCPC anatomical geometries. 25,31 A no-slip boundary condition was imposed on the rigid model walls. The time-averaged dissipation power inside the DIPI was calculated using the control volume approach, neglecting the heat losses and rate of work done by the model walls,Ė where T, p, u, Q, ρ, andn refer to the period of flow cycle, static pressure, velocity, flow rate, density, and surface normal at the boundaries (CS), respectively. Single harmonic and multi-harmonic caval waveforms were discretized into 48 and 100 time steps, respectively. The residual convergence criterion for the continuity equation was considered to be smaller than 10 −6 in all time steps. In addition to the residual convergence criterion, we monitored the outlet flow of the system to ensure that it reached a steady state. This was carried out for both steady and pulsatile inlet flows (see Sec. IV of the supplementary material for detailed information on our convergence analysis). The first three flow cycles were excluded from the analysis in order to eliminate any possible start-up effects.

B. Outlet boundary condition
To address issues associated with mass flow convergence due to backflow within patient-specific arterial and venous flow waveforms, a pulsatile outflow mass-split boundary condition was introduced to assign the flow at multiple outlets. An iterative sub-routine, incorporating a pressure boundary condition derived from the Bernoulli equation, was assigned at each outlet. The operational rationale here is to establish total energy at each outlet and distribute both static and dynamic parts of the energy in order to satisfy the desired flow rate distribution within the outlets. Systematic verification tests and characterization of this new outlet boundary condition are comprehensively described in Secs. III and IV of the supplementary material. In addition, by incrementally increasing the length of the pulmonary arteries, a set of auxiliary verification runs was performed where the flow profiles at different outflow cross sections and at the DIPI were recorded. At an extension of 11 vessel diameters, a fully developed outlet velocity profile was achieved, and more importantly, no influence on the internal TCPC flow field was observed.

C. Inflow waveforms for pulsatility analysis
Multi-harmonic periodic inflow conditions [Q 1 (t) and Q 2 (t)] were specified at the inlet boundaries of the 1DO DIPI model (inlet-1 and inlet-2, respectively) to investigate the effect of inflow pulsatility on power loss, In the above equations, "j" is the imaginary unit, the subscript "i" refers to each harmonic, and N denotes the total number of harmonics defining the waveform. The trigonometric functions are expressed in the form of complex exponentials in order to facilitate the derivation of power loss formulations. The amplitude (A i1 , A i2 ) and frequency ( f ) of the inflow oscillations and the phase angle between waveforms (Φi) were identified as the main variables that affect the pulsatile DIPI power loss. To be compatible with the baseline flow conditions of our earlier studies 25,52 and compare the computed power loss values, all inlet flow waveforms were scaled to provide 4 l/min of the total inlet flowrate through the conduit (corresponding to a typical pediatric cardiac output, flow regime). The ratio of steady inflow component of A 01 to A 02 was adjusted by the inflow flow split parameter (β), which defines the split of total flowrate between the inlets. To illustrate non-dimensional oscillation amplitudes, α i1 and α i2 represented the maximum variation of Q 1 and Q 2 from the corresponding steady flow components (A 01 and A 02 ), respectively. To investigate the isolated effect of each pulsatility variable, simulations were performed using single harmonic waveforms and the results were extended to multi-harmonic cases, as demonstrated in Sec. IV. To simplify the notation, Φ, α 1 , α 2 , A 1 , and A 2 is used to refer to the inflow pulsatility parameters associated with the single harmonic pulsatility analysis. The effect of the inflow phase shift  (24) was investigated by varying the phase angle at 15 ○ increments until Q 1 led Q 2 by a complete flow cycle. This procedure was performed for different α 1 values (0.3, 0.5, 0.8, and 1.0) in order to identify the dependency between the effect of the phase shift and the amplitude of inflow oscillations on power loss. The effect of waveform frequency was investigated by conducting simulations within both physiological (f = 0.3 Hz, 1 Hz, 1.8 Hz) and extreme cases (0.01 Hz and 5 Hz), having in-phase inflow waveforms with α 1 = 1.0. These frequency analyses (at each f ) were also repeated for waveforms with an arbitrary phase shift of 180 ○ and also with arbitrary Q 1 amplitudes (α 1 = 0.3 and α 1 = 0.5). These simulations were performed in order to investigate any possible "crosstalk" between the frequencydependent power loss variations and the effect of other pulsatility parameters on power loss. For all single harmonic pulsatility analysis, α 2 was fixed to 0.3 and β was taken as 1.5. A summary of the simulations performed for the pulsatility characterization analysis is given in Table I.

D. Identification of inflow indices based on quasi-steady flow theory
Based on our previous study, 33 the laminar flow with the Poiseuille-like velocity profile was maintained within the inflow branches until close proximity to the junction zone [see Fig. 1 For these uniform flow regimes, the estimation of frictional losses based on the quasi-steady flow assumption is regarded a reasonable analytical approach. 53 This assumption forms the basis for the following simple power loss formula (the details of this derivation are provided in Sec. I of the supplementary material): where the subscript "i" refers to the inflow segments 1 and 2. The pulsatile flow exponent (γ) can typically be valued as 2 based on the well-known Darcy-Weisbach equation. 54 The lumped resistance of the inlet pipe segments (R -and alternatively γ) can be calculated through numerical or experimental methods. 37 Energy dissipation in the junction zone and the downstream outflow branches is more complex and depends on the interaction of both inflow streams in the junction zone, where energy dissipation is nonlinearly dependent on the instantaneous total inflow Similarly, for outflow branch regions, the fraction of the total inflow directed to each outlet (i.e., governed by the outflow split coefficient OFS = Q outflow_1 /Q in ) dictates power loss through a similar power law expression as in Eq. (6). Instantaneous power loss is expected to vary within these regions since Q in (t) varies significantly with the phase difference Φ (Fig. 2).
Substituting inflow waveforms [Eqs. (2) and (3) into Eq. (7)] and evaluating the time-averaged power loss inside each outflow branch and the junction section show that Φ i alters the power loss within the branches and junction based on the nonlinear relationship between the power loss and the total flow rate [see γ ≥ 2 in Eq. (2)]. 37 In contrast, the time-averaged power loss within the inlet branches is independent of the caval inflow phase difference Φ i and will vary only with the amplitude of the flow oscillations (A i1 , A i2 ) (refer to Sec. II of the supplementary material for the relevant derivations).

FIG. 2.
Phase angle Φ between the two inflow waveforms affects the pulsatility of total inflow [i.e., Q in (t) = Q Inflow-1 (t) + Q inflow-2 (t)] and alters the amplitude of the net inflow fluctuations from the steady (non-pulsatile) inflow. The fluctuations are maximum when the inflow waveforms are out-of-phase (Φ = 180 ○ ) and minimum for in-phase configuration (Φ = 0 ○ , 360 ○ ).

ARTICLE scitation.org/journal/adv
Based on the proposed relationship between the region-specific pulsatility and power loss, three indices were introduced to quantify pulsatility within the 1DO DIPI geometry. The absolute value of the time-averaged fluctuations of Q in (t) relative to Q steady quantifies the amount of pulsatility within the junction zone and the outlet branches. Incorporating Eq. (2) to (7), the Total Inflow Pulsatility Index (TIPI) is defined as where )dt and T refers to the period of the flow cycle. Likewise, the pulsatility of the inlet branches was quantified using the indices named as Inflow Pulsatility Indices (IPIs), where the subscript k counts for inflow branches 1 and 2.

E. Analytical relationship between conduit power loss and pulsatility
As shown in Fig. 1(a), the DIPI flow pathway was discretized into three distinct regions: inflow branches, outflow branches, and the junction zone. Each region has different pulsatile characteristics contributing to the flow physics and, therefore, responds differently to variations in inflow pulsatility (i.e., changes in A i1 , A i2 , and Φ). Based on the limited turbulent mixing within the junction zone and uniform distribution of the inflow to each outflow branch (α = 0.5), flow within the junction section and outflow branches is assumed to behave similarly. Hence, the unsteady DIPI power loss (PLDIPI) was calculated as the summation of regional power loss at the inflow branches and the lumped outflow-junction zone by incorporating Eq. (6), where the subscript "i" refers to each section: inflow branches (i = 1, 2) and lumped outflow-junction (i = 3). The time integral was evaluated by substituting the corresponding flow rates Q 1 , Q 2 , and Q 3 = Q in for Secs. I-III, respectively. Regional pulsatility indices identified in Sec. II D [i.e., Eqs. (8) and (9)] were substituted into Eq. (10) to identify the non-dimensional functional form of PLDIPI in terms of the inflow pulsatility and steady DIPI power loss (PL steady ), where C 1 , C 2 , and C 3 are the constants derived from the lumped hydraulic resistances of Secs. I-III, respectively. Non-dimensional formulation identified the influence of the relative steady inflow ratio (β) on the energy dissipation due to the inflow fluctuations within the inlet branches. For the DIPI model, the resistance of inflow branches was taken as identical, C 1 = C 2 (see Sec. II of the supplementary material for details of this derivation). Statistical analysis to assess the strength and significance of the correlations was performed with the MedCalc software (Ostend, Belgium). Correlations were described with the use of scatter plots, such as the linear regression fit and Spearman correlation coefficients in the absence of standard normal distribution.

III. RESULTS
A. Effect of amplitude and phase angle dependent inflow pulsatility on unsteady DIPI power loss 1DO DIPI power loss varied incrementally with phase angle (Φ) variation between 0 and 360 ○ and followed a characteristic bell-shaped (cosine) curve (Fig. 3) for all oscillation amplitudes α 1 (0.3, 0.5, 0.8, 1). Maximum and minimum power losses appeared when waveforms were in-phase (Φ = 0) and out-of-phase (Φ = 180), respectively. The power loss change between the two extremes of Φ (180 ○ with respect to 0 ○ ) was calculated as 2.5%, 6.1%, 7.3%, and 8.6% for α 1 0.3, 0.5, 0.8, and 1.0, respectively. Hence, the change of maximum power loss due to Φ variation increased as α 1 increased even though the cosine trend between the power loss and Φ was preserved.
Φ and α 1 dependent power loss variation was further investigated by examining the pulsatility indices, TIPI and IPI.

ARTICLE
scitation.org/journal/adv configurations (Φ = 0, 360 ○ ) at all amplitude oscillations, α 1 = 0.3, 0.5, 0.8, 1, respectively. Examining this relationship for the interim phase angles, TIPI correlated positively with power loss and successfully traced the non-linear power loss variation for different Φ. For the entire dataset, the strength of the association between TIPI and power loss was measured using Spearman's correlation coefficients (ρ = 0.96, p < 0.0001). Based on this positive correlation, the global minima of energy dissipation were realized when the inlet waveforms were non-pulsatile (i.e., steady inflow condition, TIPI = 0). The pulsatile component of power loss was quantified using normalized pulsatile power loss (NPPL), which is the ratio of total power loss over the steady-state (NPPL = PL DIPI PL steady ). It is worth noting that the total flowrate was identical for both steady and pulsatile flow conditions. Figure 4 shows the Φ-dependent power loss variation with respect to TIPI 2 for each amplitude α 1 . Power loss changed along four different linear curves with identical slopes (n = 1.01 ± 0.07), which were associated with a particular α 1 value. Strong linear correlation (R 2 = 0.99) was found between the NPPL and TIPI 2 based on the Φ-dependent power loss variation for each α 1 . This consistent agreement indicates the utility of using TIPI as a pulsatility index to represent power loss variation due to Φ-dependent inflow pulsatility changes. Figure 4 further illustrates that multiple power loss values corresponded to the same TIPI. Therefore, TIPI alone was insufficient to fully characterize the pulsatile 1DO DIPI power loss. The regression curve in Fig. 5 shows that a linear relationship between IPI 2 variation and power loss was also identified (n = 2.1, R 2 = 0.98) for α 1 -dependant power loss variation at the same TIPI.  To validate this piece-wise power loss estimation approach, we calculated power loss using control volumes surrounding each of the inflow-1, inflow-2, and lumped outflow-junction sections for arbitrary Φ and α 1 values. As shown in Table II, Φ led to power loss variation only within the outlet-junction zones. In contrast, α 1 altered the power loss at all branches except for the inflow-2. Power loss inside the inflow-1 branch expressed a linear correlation with IPI 1 2 regardless of Φ. Likewise, Φ-dependent power loss variation within the outflow-junction section correlated linearly with TIPI 2 for an arbitrary α 1 . These results prove the utility of IPI 2 and TIPI 2 to fully describe the relationship between the inflow pulsatility and power loss variation within the associated sections and hence TABLE II. Regional power loss variation. Regional power losses (mW) evaluation at the inflow branches and lumped outflow junction section (the geometry is shown in Fig. 1) and correlation with pulsatility indices based on the control volume method. Φ, α 1 , IPI, and TIPI refer to inflow phase angle, non-dimensional flow amplitude oscillations, inflow pulsatility index, and total inflow pulsatility index, respectively. α 1 -variation simulations were performed at α 2 = 0.3, Φ = 0 ○ , and β = 1.5, whereas Φ-variation simulations were performed at α 1 = 0.1, α 2 = 0.3, and β = 1.5.

B. Effect of frequency dependent inflow pulsatility on unsteady DIPI power loss
The effect of inflow oscillation frequency on 1DO DIPI power loss was investigated in detail by evaluating multiple in-phase inflow waveforms with different frequencies ranging from 0.03 Hz to 5 Hz. Associated Womersley numbers (Wo = D 2 √ wρ μ ) ranged between 1.5 and 20, whereby D, w, ρ, and μ refer to the conduit diameter, angular frequency, density, and dynamic viscosity of the fluid, respectively. To enable comparison with the previous simulations at f = 1 Hz, f -dependent power loss variations were reported with respect to the baseline power loss at f = 1 Hz. It is significant to note that normalized power loss decreased remarkably by 10% with the increase in inflow frequency, following a sigmoidal relationship, as shown in Fig. 6.
In addition to the rapid changes observed between Wo = 6 and 13, power loss converged asymptotically to maximum (PLmax) and minimum (PL min ) at Wo min = 3 and Womax = 15, respectively. A Boltzmann four-parametric sigmoid function (R 2 = 0.99) sufficiently described this trend, where Woavg = 9.8 denotes the Womersley number, which corresponds to the power loss halfway between the maximum and minimum. The term "Slope" is used to describe the steepness of the sigmoid curve and was calculated as 1.751. Frequency simulations performed with Φ = 180 datasets yielded excellent agreement with the above results (9% variation between 1 Hz and 5 Hz), indicating the independence of f -dependent power loss variation from the Φ-dependent effects on DIPI power loss. In contrast, the frequency variations for α 1 = 0.5 yielded relatively lower power loss variation (6%) between the high and low Wo number ranges. The frequencydependency analysis was further investigated at α 1 = 0.3, which yielded even smaller power loss variations (∼2%) with frequency, as shown in Fig. 7.
C. Non-dimensional analytical expression to estimate unsteady power loss and its predictive capability Based on the quasi-steady flow assumption, an analytical expression to predict pulsatile power loss changes was developed to complement the steady TCPC energy dissipation formulation. 37 Considering the well-known limitations of the quasi-steady flow approach for highly transient flows, 53,55 the derived analytical formula is designed to accurately represent the pulsatile response of the 1DO DIPI geometry and identify the functional relationship between the 1DO DIPI power loss and pulsatility indices (PL ∼ IPI 2 , TIPI 2 ). Combining the correlation constants identified for each independent pulsatility index (C 1 = 2.12, C 3 = 1.11), the final form of the pulsatile 1DO DIPI power loss equation was expressed as

FIG. 7.
The non-linear relationship between the steady hydraulic resistance, CO, and inflow split parameter (β = A 01 /A 02 ) indicates increased steady power loss (PL steady = A 0 2 Co) for unbalanced steady inflows (β ≠ 1). Based on the equal vascular resistance assumption (R 1 = R 2 = R 3 = 1), CO converges to a phenomenological resistance value of 2 as β approaches to zero or infinity. Note that for different branch resistances (R 1 = R 2 = R 3 /2 = 1), the CO vs β relationship remains the same, but CO converges to a different asymptotical value of 1.5. A 01 and A 02 refer to the steady component of the inflow waveforms and A 0 = A 01 + A 02 .

ARTICLE
scitation.org/journal/adv Utilizing the orthogonality principle of trigonometric functions, this relationship enables the pulsatile power loss analysis of any multi-harmonic arbitrary inflow waveform inside the idealized 1DO DIPI model. IPI 1,2-adjusted refers to the inflow pulsatility index adjusted by the unbalanced steady inflows (β), as shown in Sec. II of the supplementary material. The predictive accuracy of Eq. (13) was demonstrated by comparing the power loss calculated by the analytical formula and the associated CFD solution for four multi-harmonic arbitrary inflow waveforms. As shown in Fig. 7, each waveform incorporated different degrees of inflow pulsatility, whereas the total steady flow was identical (i.e., to 4 l/min). Table III shows the inflow pulsatility of each waveform represented by the proposed pulsatility indices.
Frequency content of the inflow waveforms was quantified by the Inflow Frequency Index (IFI), which averages the amplitude weighted frequency of the major harmonics of the two inflow waveforms. The number of harmonics utilized in the IFI calculation was based on the desired accuracy of the waveform reconstruction (R 2 = 0.90), TABLE IV. Predictive capability of the analytical power loss equation. The predictive capability of the analytical power loss equation was compared to the numerical power loss calculated by the computational fluid dynamic (CFD) evaluation of four multiharmonic waveforms inside one diameter offset double inlet pipe intersection (1DO DIPI) geometry. % |Error| shows the difference between the CFD power loss and the analytical estimate. CFD-NPPL and analytical-NPPL refer to the pulsatile power loss normalized by the corresponding non-pulsatile (steady power loss) calculated by the CFD solver and the analytical power loss equation (13), respectively. Four multiharmonic inflow waveform sets were scaled to the same total inflow rate (3 l/min) and denoted as W1, W2, W3, and W4. In the above formula, f i and A i refer to the frequency and amplitude of the harmonics associated with inflow waveforms 1 and 2.
In order to preserve the non-dimensional form of the formulation, IFI was normalized and expressed in terms of the Womersley number. Table IV and Fig. 8 show the comparison between the analytically predicted and CFD-derived actual power loss for each waveform set. Consistent agreement (Pearson's linear correlation coefficients, p = 0.96, p < 0.04) between the CFD solution and analytical calculation indicates that the proposed analytical power loss equation successfully predicted the relative pulsatile energy efficiency of each inflow scenario. In addition, predicted power loss values estimated the actual power loss within 13% error margin and confirmed the utility of frequency-coupled quasi-steady theory to estimate the pulsatile 1DO DIPI power loss.

A. Role of inflow pulsatility on double inlet pipe intersection power loss
The present study demonstrates the significant impact of inflow pulsatility on energy dissipation in non-impinging stable crossflow jets. Adopting the physiological settings of typical 1DO TCPC geometry, we were able to characterize the effect of each inflow pulsatility parameter (α, Φ, f) on the power loss. Our research results indicate that by increasing the amplitude oscillation of one of the inflow waveforms (α 1 = 0.3 to 1.0), power loss can also increase nonlinearly up to 60% from the non-pulsatile baseline, which is quite significant for any hydrodynamic application. As demonstrated in an earlier study, 33 modulation of the inflow phase angle results in 2%-8% power loss variation inside the 1DO DIPI geometry, depending on the amplitude of the inflow oscillations. The waveform frequency also showed a notable impact on power loss (2%-10%), which was dependent on the pulsatility amplitude.
Our previous analysis 33 indicated that the variation of power loss with changes in the phase angle correlates with the strength of the rotational vortices within the junction zone and outlet branches. In this study, we quantitatively demonstrated that Φ-variation altered the time-averaged power loss within the junction and outflow branches. For surgically created vascular pipe junctions, the variation of power loss due to phase shifts is strongly related to the hemodynamic response within the TCPC junction and pulmonary arteries, which depend nonlinearly on the total incoming cardiac output (CO) and the fraction of CO directed to each lung. Since the instantaneous total operating CO varies significantly with the phase difference (see Fig. 2), the corresponding mean power loss also varies within these regions. To summarize, the TIPI, as an index to quantify the pulsatility of total inflow [see Eq. (8)], quantified the degree to which power loss within the junction zone and outlet branches was influenced by the inflow phase-angle as well as the amplitude of inflow oscillations. Similarly, the IPI [see Eq. (9)] accurately evaluated the oscillation amplitude-dependent power loss variation within the inflow branches.
Detailed analysis of the DIPI flow field indicated that for low frequencies (∼Wo min ), the flow and pressure waveforms had almost an in-phase configuration and the flow profile displayed a Poiseuille-like parabolic appearance throughout the inflow and outflow branches. In contrast, for high frequency oscillations (∼Womax), the flow profile was not able to reach a fully developed shape and yielded lower shear stress and consequently lower power dissipation. Furthermore, the peak flow was not able to keep pace with the peak junction pressure drop (i.e., phase lag), which also lowered the power loss compared to the low frequency cases. It is also important to note that the flow settings of the 1DO DIPI model represented an inverse Womersley problem, whereby the inflow oscillations resulted in the fluid motion, as opposed to the transient pressure gradient oscillations described by the well-known Womersley flow theory. 56 We also found that the effect of f -related power loss variation depends on the oscillation amplitude. The power loss variation between Wo ∼ 2 and 20 was calculated at 5% and 2% for IVC flow waveforms with 0.5 and 0.3 oscillation amplitudes, respectively. These response results demonstrate strong agreement with the Reynolds number dependence of flow reversal based on the in vitro experiments performed inside a bench-top compliant aortic flow loop. 57 For low Reynolds numbers, the Stokes layer flow instabilities are minimized due to the high frequency pressure oscillations. In contrast, the flow reversal is increased even for smaller Wo, when Re is larger within the aorta (a straight internal pipe segment).

B. Generalized pulsatile power loss with multiple inlets and outlets
Dasi et al. 37 introduced non-dimensional metrics to characterize TCPC power losses with steady caval inflow and highlighted the total caval flow rate (equivalent to the CO for balanced pulmonary and systemic flows) as a major parameter for determining the degree of power loss. 37,57 Expanding on these studies, our present pulsatile investigation indicates that for a fixed, time-averaged physiological total caval flow rate, the mean energy loss can be altered significantly as the pulsatile content of the caval flows (oscillation amplitude, frequency, and phase-angle between caval flows) changes. The analytical expression, which incorporates the quasi-steady flow theory and the clinically relevant pulsatility metrics introduced in this study, further complements the steady TCPC power dissipation formulation 37 and enables the prediction of pulsatile power loss changes without time-demanding CFD calculations. For the selected cavopulmonary waveform sets, the predicted vs calculated power loss ranged within a 13% error margin and confirmed the utility of the proposed frequency-coupled quasi-steady flow theory, incorporating pulsatility indices. This methodology can be easily interfaced with any real-time imaging modality (Doppler ultrasound or MRI), which would be particularly useful for comparing the pulsatile performance of venous flows for any arbitrary Fontan patient for immediate feedback in the clinic.
The proposed pulsatile power loss characterization method can be further generalized for any 1DO DIPI geometry with arbitrarysized conduit dimensions, provided the correlation constants in Eq. (10) are identified beforehand for each junction topology template. The present study shows that due to the linear relationship between PL and both TIPI 2 and IPI 2 , a reduced number of operating points can be incorporated for this task, as opposed to the comprehensive and laborious system characterization method employed in this study. Correlation constants for TIPI and IPI can be identified by conducting only three CFD runs, where TIPI and IPI will be altered one-at-a-time from the baseline operating point. In addition, two more simulations at relatively higher and lower frequency contents (i.e., around Wo ∼ 15 and Wo ∼ 1) are required to identify the sigmoid shaped f-dependent power loss curve. Hence, a minimum of five operating points would be sufficient for characterizing the pulsatile power loss response of any 1DO DIPI geometry with arbitrarily sized conduit caliber and complex 3D topology. Moreover, a higher number of operating points would improve the accuracy of the regression analysis and the predictive capability of the analytic expression further.
It is worthwhile to note that the analytical formulation pioneered in this study clearly reveals the effect of unbalanced steady inflows (β) on both the steady and pulsatile DIPI power loss. When two inflow waveforms have identical non-dimensional inflow oscillations [α 1 = α 2 (IPI)], Eq. (S9) shows that the waveform with the higher steady flow component dissipates more pulsatile

ARTICLE
scitation.org/journal/adv power within the inflow branches. In comparison, although the numerical simulations of Khunatorn et al. 58 demonstrated the increase in steady power loss due to unbalanced inflows, the theoretical foundation behind these findings was unclear. Incorporating the steady-state 1DO DIPI power loss expression given by Eq. (S7) (PL steady = A 0 2 Co) and assuming equal vascular resistance along each branch (R 1 = R 2 = R 3) , the hydraulic resistance term demonstrates the parabolic relationship between the steady power loss and β, which, in turn, complements the aforementioned numerical findings. 58 It is important to note that the steady-state power loss depends strongly on the relative resistance of the inlet vessels. Hence, unbalanced inflow split conditions can actually lower the power losses when the venous flow split transverses through the caval branch with increasing lower resistance.

C. Clinical implications of flow pulsatility for patients with single ventricle
The significant effect of caval pulsatility on power loss in the 1DO DIPI model demonstrated in this study may provide the impetus to modulate the pulsatile content (i.e., phase angle and amplitude) of inflow waveforms when the energy efficiency of the connection is a major concern. In particular, our numerical methodology can be utilized to analyze and optimize the hemodynamics efficiency of pulsatile flow waveforms generated by emerging mechanical assist therapies, such as venous assist devices 59,60 and enhanced external counter pulsation, 61 for single ventricle patients. These research findings can lead to the design of optimized mechanical support that can not only boost global hemodynamics performance (i.e., higher diastolic venous flow, higher preload, and higher CO) but also modulate the inferior caval flow to recover the physiological hemodynamic state within the SV circuit for patients with heart defects.

V. LIMITATIONS
Lumped representation of the frequency content of the inflow waveforms neglected the interactions between different harmonics and assumed that the power loss response of the multi-harmonic waveforms can be represented with an average single harmonic frequency. This effect is significant for junctions with no inlet offset. In addition, the utility of mean frequency content of the two inflow waveforms assumed that the frequency content within these two inflow branches were inherently similar, which was essentially true for the flows investigated. As the f -dependent power loss response within the DIPI geometry appeared non-linear (Fig. 7), the proposed definition of IFI will reduce the accuracy of the power loss prediction for inflow waveforms with non-matching frequency content, which can be improved further. In addition, the amplitude dependency of the frequency response was not investigated for larger caval oscillations (α 1 > 1) and not included for the formulation used in this study. As such, this expression is valid only under the laminar flow regime within the tested Reynolds number range (Re ∼ 600 to 2000). Deviations from the actual power loss should be expected when turbulent mixing and transitionary flow conditions are present.
For the simplicity of the given derivations, the hydraulic resistance of each inflow branch was approximated as one half of the resistance of the lumped outflow-junction section (R 1 = R 2 = R 3 /2). This assumption strongly regulated the relative contribution of IPI 2 and TIPI 2 terms to the overall power loss estimation. Therefore, the predictive accuracy of the formulation may be further improved by incorporating the actual resistance of the branches. The assessment of actual resistance may be critical if the DIPI geometry diverges from the symmetrical shape and incorporates non-identical inflow and outflow branches. In addition, the proposed non-dimensional pulsatile formulation can be further revised to include the OFS coefficient for non-uniform pulmonary (outflow branch) flow split scenarios.
In order not to complicate the analytical form of the power loss equation, the cardiac output (CO), which depends on β, was treated as a constant. The associated error introduced by this assumption was estimated by re-evaluating the power loss for the aforementioned multi-harmonic waveforms using variable CO (β). It was shown that CO changes within a small range (0.7 and 0.8) and is otherwise almost constant ∼0.79 for 0.6 ≤ β ≤ 1.5. Hence, CO affected the associated power loss calculations less than 1% except for the W4 waveform, which was 4% due to the fact that β approached ∼0.2.

VI. CONCLUSION
The parametric analytical formulation pioneered in this study introduced easy-to-calculate pulsatility indices that correlate with the pulsatile energy efficiency of double inlet pipe intersection (DIPI) geometry with limited cross-flow mixing. Our results indicate that the power loss within the inflow branches, the lumped outflow-junction section, and the whole conduit correlates strongly with the pulsatility of each inflow waveform (IPI, p < 0.05), the total inflow pulsatility (TIPI, p < 0.05), and the inflow frequency content (IFI, p < 0.05), respectively. These findings suggest that provided the total flow rate of the junction is fixed, the inlet waveform set with sharp local peaks and low-frequency flow oscillations will generate higher power loss inside the 1DO conduit in comparison to the high frequency low amplitude inflow pulsations. The proposed frequency-coupled quasi-steady power loss vs waveform pulsatility relationship holds for the 1DO DIPI model and illustrates the power loss characteristics of the caval flow waveforms for this particular geometry. Our results may have important implications for the patient-to-patient comparison of surgical connections since similar parametric studies can be employed to characterize the pulsatile response of arbitrary patient-specific surgical templates to arbitrary inflow waveforms as well as industrial cross-flow pipe junctions. Coupling the analytical pulsatile power loss expression with the previous non-dimensional steady power loss formulation 37 provides a valuable predictive tool to estimate the pulsatile energy dissipation for any arbitrary junction topology and flow waveform without the use of costly CFD computations.

SUPPLEMENTARY MATERIAL
See the supplementary material for detailed information on the derivation of pulsatile and steady power loss formulations, validation of outlet boundary conditions used in this study, and grid sensitivity analysis.