Three-dimensional Theoretical Model Research Articles

Three-dimensional numerical simulation and theoretical model of a hollow droplet impacting on a solid surface

The phenomenon of a hollow droplet impacting on a solid surface is widely found in various fields. The dynamic characteristics of hollow diesel droplets impacting on a solid surface are studied by combining numerical simulation and theoretical analysis. The dynamic contact angle model presented in this paper couples the advantages of existing dynamic contact angle models for simulating both the spreading and retracting stages. It also considers the continuous variation of the contact angle during the maximum spreading state. Compared to existing models, the maximum error has been reduced from 14.9% to 4.6%. The effects of impact velocity, impact angle, and volume ratio of a hollow droplet on the spreading and jetting characteristics are investigated by three-dimensional numerical simulations. It is found that air entrainment occurred in the counter-jet, and the presence of the impact angle increased the asymmetry of the counter-jet and spreading liquid film, promoting fingerlike splashing at the front liquid film. Based on energy conservation law, the theoretical prediction models of the maximum spreading coefficient of the hollow droplet impacting on the surface and the velocity of the counter-jet at the maximum spreading state are established using the multi-regional modeling method and the energy distribution principle. Compared with existing hollow droplet theoretical models, the proposed theoretical models exhibit a more concise expression, higher accuracy, and wider applicability range.

Attributes influencing guests’ experiences: a comparison of nonprofit and for-profit peer-to-peer accommodation platforms

Peer-to-peer accommodation has generated an ecosystem of platforms with different business models (i.e., for-profit and nonprofit). This study aims to identify and compare attributes that influence guests’ experiences as reviewed on the for-profit platform Airbnb and the nonprofit platforms Couchsurfing.com and HomeExchange.com according to a three-dimensional experience theoretical model and a methodological approach to interpret these attributes. The study used text-mining techniques to analyze 772,768 online travel reviews representing Spain’s four most-visited cities. Findings show that attributes influencing guests’ experiences in the case of nonprofit platforms relate to the authenticity dimension of experience (e.g., existential values and travel philosophy). Furthermore, guests reported that their guest–host interaction was the most representative attribute and that, unlike with Airbnb, such interaction helped to create a more authentic experience. By contrast, attributes of guests’ experiences in the case of for-profit platforms related to the physical amenities and characteristics that guests would expect to find in hotels. Those results can allow destination managers and accommodation practitioners to better understand users of peer-to-peer accommodations and thereby design more suitable strategies and experiences for them.

Three-dimensional theoretical model for effectively describing the effect of craniomaxillofacial structural factors on loading situation in the temporomandibular joint

BackgroundTemporomandibular joint (TMJ) overloading is considered a primary cause of temporomandibular joint disorders (TMD). Accordingly, craniomaxillofacial structural parameters affect the loading situation in the TMJ. However, no effective method exists for quantitatively measuring the loading variation in human TMJs. Clinical statistics, which draws from general rules from large amounts of clinical data, cannot entry for exploring the underlying biomechanical mechanism in craniomaxillofacial system. The finite element method (FEM) is an effective tool for analyze the stress and load on TMJs for several cases in a short period of time; however, it is difficult to generalize general patterns through calculations between different cases due to the different geometric characteristics and occlusal contacts between each case. Methods(1) This study included 88 subjects with 176 unilateral data to measure angle (α) of the distance to the plane of occlusion. The bone destruction score was evaluated for clinical statistics. To rule out effects of the potential factors and ensure the generality of the study, one participant with no obvious bone destruction was selected as the standard case for establishing the three-dimensional (3D) theoretical model and FEM. (2) Three groups of forces, including biting, muscles and joint reaction forces on mandible, were adopted to establish a 3D theoretical model. (3) By modifying the sagittal α and coronal three types of deviation angle (φ) of the original model, nine candidate models were obtained for the FEM studies. Results(1) The static equilibrium equations, were used to establish a 3D theoretical model for describing the loading of the TMJ. The theoretical model was validated by monotonously modifying the structural parameter in comparison to two-dimensional theoretical models reported previously; (2) The force on the TMJ gradually decreased with α, and this trend was validated by both clinic statistics and FEM results; (3) The effects of the three types of deviation angle were different. The results of the case where only rotating biting forces were considered was consistent with clinical statistics, indicating that the side with lower α experiences higher TMJ load. (4) Changing the unilateral proportionality coefficients of biting and muscle force produced opposite effects, wherein the effects of the muscle force were stronger than those of the biting forces. ConclusionsA negative correlation was observed between the joint load and α. Among the three types of asymmetric deformities, occlusal deviations were the primary factors leading to TMD. Unilateral occlusion can result in a greater load on the ipsilateral joint and should be avoided when using the side corresponding to the TMD. This study provides a theoretical basis for the biomechanical mechanism of TMD and also enables the targeted mitigation and treatment of TMD through structural modification.

Theoretical study of influence of laser pulse chirp on terahertz emission characteristics of gas induced by two-color laser field

With the development of terahertz (THz) wave research, the demand for controllable THz sources is increasing. How to obtain the regulated THz waves has been one of the research hotspots and key problem in the field of THz science. There have been researches in which the resulting THz wave is modulated by changing the wavelength, relative phase, energy, or chirp of the laser produced by a two-color laser. In this work, we establish a three-dimensional theoretical model of THz wave generation and subsequent propagation induced by two-color laser. And we investigate the influence of chirp modulation of different laser on THz wave by chirp modulation of the fundamental wave (FW) and the second harmonic wave (SHW) of two-color laser, including THz wave amplitude, THz wave center frequency and spectrum width, and analyze the physical mechanism of related phenomena. At the same time, the effects of different orders of magnitudes of laser chirp parameters (femtosecond and picosecond) and initial phase of laser pulse on THz wave parameters are also studied. The results are shown below. 1) In the two-color laser, the chirp of FW mainly affects the shape of THz wave when the initial phase is unchanged. The chirp modulation of SHW can cause the amplitude of THz wave to change significantly, and affect the center frequency and spectrum width of THz waves. 2) In the case of laser pulse width of femtosecond order, 40 fs is taken as an example. When the chirp parameter is of femtosecond magnitude, the chirp parameter has a great influence on the THz wave generation efficiency of two-color laser filament. At the picosecond magnitude, the chirp parameter has a weak effect on the THz wave energy and mainly affects the phase of the THz wave. 3) The initial phase of the two-color laser can aid in chirp modulation of THz wave to optimize the energy generated. 4) The initial phase of two-color laser can assist in the process of chirped laser modulation of terahertz waves to optimize the energy generated. Our research shows that the chirp modulation, as a controllable parameter, has multiple regulation effect on the properties of radiated THz waves. The related research results provide a new idea and basis for studying the generation and regulation of THz waves.

TriboNano shield – Scalable manufacturing anti-smog multi-level structured nanofiber air filter with co-enhanced purification performance towards self-powered window screen

Nanofiber air filter has garnered tremendous attention due to large specific surface area, large porosity, highly interconnected aperture channels, etc. However, its unitary interlayer structural construction encounters limitation of filtration performance with a low yield. Here, we architect a triboelectric-induced multi-level interlaminar composite structure-based nanofiber window screen (T-MLNS), manufactured continuously utilizing free surface electrospinning. Subjoined with triboelectric charges (active type), T-MLNS possesses co-enhanced adsorption capacity of aerosols. Additionally, MLNS is constructed with finer fibrous layers for precise filtration and coarser fibrous layers in the middle facilitating airflow passing, under physical filtration mechanisms (passive type). Significantly, three-dimensional theoretical models are innovatively employed with numerical simulation findings that multi-level structures present superb filtration performance than uni-structure ones. In the synergistic effect of the active and passive typed mechanisms, T-MLNS is prolonged service life for precise capture of ultrafine aerosols (Diameter: 0.26 µm) with enhanced filtration efficiency up to 99.9%. The integrated multi-zone structure design creates a pattern-shape fiber mesh, resulting in prominent air permeability (∼100 mm s−1), optical light transmission (> 60%), and robust mechanical properties. This combination of active and passive filtering self-powered window screens offers a unique concept for revolutionising interior air quality in industrialization.

Three-dimensional dynamics of supported pipe conveying fluid with arbitrary initial spatial shape

In order to meet different requirements of engineering applications, pipes conveying fluid usually have the geometric characteristic of a spatial initial shape. In this paper, a three-dimensional theoretical model based on absolute node coordinate formulation (ANCF) is proposed to analyze the dynamical behavior of spatial pipes conveying fluid with arbitrary initial shape. By introducing the spatial Frenet frame as the local coordinate, the global coordinate of each point on the pipe centerline is intuitively represented first. Considering the tensile, torsional and bending deformations of the pipe, the governing equations of spatial pipes conveying fluid are derived by means of the generalized Lagrange equation. Then the process of solving nonlinear static deformation, natural frequencies and nonlinear dynamic responses of the pipe is given. Further, by comparing the results calculated by the ANCF method with those of previous studies on straight and curved pipes, the effectiveness of the proposed three-dimensional ANCF model is verified. Finally, the nonlinear responses and natural frequencies of spatial pipes with three different initial shapes are calculated and some of them are compared with the results obtained using finite element simulations. The results show that the three-dimensional ANCF model proposed in this study has good feasibility and wide application prospect in nonlinear dynamics analysis of fluid-conveying pipes with complex initial shapes.

Theoretical, CFD modelling and experimental investigation of a four-intersecting-vane rotary expander

Expansion devices significantly impact the performance of the Vapor Compression Refrigeration system and Organic Rankine Cycles. Its improvement has been identified as one of the most crucial parts of future studies. The experiments were carried out using an improved experimental rig with direct coupling of the prototype and the dynamometer. This is done to correct any misalignments caused by external forces. The internal pressure in the working chambers was measured using six pressure transducers at various locations on the expander. A three-dimensional computational fluid dynamics and theoretical model was developed to investigate the effect of losses and inherent physical processes on the prototype. The actual geometry of the stator is taken in the computational fluid dynamics model. Conversely, employing the equation for an irregular stator shape in the theoretical model is difficult. Therefore, a circular stator is considered. The experimental results were used to validate the developed models. The prototype was tested up to 1550 rpm rotating speed, 5 bar (abs) suction pressure, and 1 bar discharge pressure (abs). The computational fluid dynamics and theoretical model results showed that though the volumetric and adiabatic efficiency was generally overpredicted, the trends were very closely predicted. The computational fluid dynamics and the theoretical model could predict the volumetric and adiabatic efficiency with a variance of <19.5% and 14.7%, respectively, for most of the experimental data points.

Effect of Bronchial Blood Flow on Respiratory Heat Exchange: A Mathematical Analysis for Infectious Diseases

Abstract The present study aims to mathematically analyze the role of bronchial blood flow on heat transfer in respiratory infections. In general, the exchange of heat transfer in various infectious diseases like COVID-19 caused by SARS-CoV-2 has adversely affected respiration by reducing the physiological efficiency of the human respiratory tract. The mechanism of heat exchange through airway walls with the bronchial blood circulation still needs to be thoroughly studied for infectious diseases. In this article, a three-dimensional (3D) spatio-temporal theoretical model is developed to estimate the possible role of bronchial blood on heat exchange during breathing. The local description of the model is presented in a comprehensive and consistent dimensionless framework to explicitly state the actual physiological background. The global description is framed by a multicompartment-based approach, and the algorithm is solved using an advanced numerical scheme to ensure computational tractability. The numerical study elucidates the role of inhalation air temperature, breathing cycles, blood perfusion rate, and mucosal hydration. The outcomes of the algorithm estimate the parameters of the isothermal saturation boundary (ISB), which is defined as the position in the respiratory tract where the temperature of inhaled air comes in equilibrium with the body core saturation temperature. The derived results help to understand the pathophysiological threshold limits and recommend the values to evaluate respiratory distress. With the variations of inspiratory flow conditions, it is observed that the ISB position shifts to the distal branches with the increment in inhalation temperature, breathing rate and virus infection, and decrement in blood perfusion rate. The two antiparallel effects are observed: inhalation of cold air transmits the viral infection, and inhalation of warm air produces thermal injury. However, both can be well controlled by suitable ventilation rates. The observed threshold values may be helpful in clinical trials to correlate the anatomic configuration with pathophysiology.

Interface and electromagnetic effects in the valley splitting of Si quantum dots

The performance and scalability of silicon spin qubits depend directly on the value of the conduction band valley splitting (VS). In this work, we investigate the influence of electromagnetic fields and the interface width on the VS of a quantum dot in a Si/SiGe heterostructure. We propose a new three-dimensional theoretical model within the effective mass theory for the calculation of the VS in such heterostructures that takes into account the concentration fluctuation at the interfaces and the lateral confinement. With this model, we predict that the electric field is an important parameter for VS engineering, since it can shift the probability distribution away from small VSs for some interface widths. We also obtain a critical softness of the interfaces in the heterostructure, above which the best option for spin qubits is to consider an interface as wide as possible.

Magnetic Charge Model for Leakage Signals from Surface Defects in Ferromagnetic Material.

A novel three-dimensional theoretical model of magnetic flux leakage (MFL) is proposed in this paper based on the magnetic dipole model. The magnetic dipole model assumes that a ferromagnetic specimen with defects is exposed to a uniform external magnetic field that causes a uniform magnetization around the defect surface. Under this assumption, the MFL can be regarded as arising from magnetic charges on the defect surface. Previous theoretical models were mostly used to analyze simple crack defects such as cylindrical and rectangular cracks. In this paper, we developed a magnetic dipole model for more complex defect shapes such as circular truncated holes, conical holes, elliptical holes, and double-curve-shaped crack holes to complement the existing defect shapes. Experimental results and comparisons with previous models demonstrate that the proposed model provides a better approximation of complex defect shapes.

Study on heat flow transfer characteristics and main influencing factors of waxy crude oil tank during storage heating process under dynamic thermal conditions

In response to the energy conservation and emission reduction targets proposed by China's Fourteenth Five-Year Plan, it is necessary for oilfield enterprises to achieve the purpose of reducing energy consumption and carbon emissions of crude oil storage under the premise of ensuring safe production of oil depots. As the preferred oil storage facility for crude oil depots, the internal heat flow transmission process of storage tanks plays a vital role in the overall energy consumption of oil depots. In this paper, a three-dimensional theoretical model of the heating process of large waxy crude oil tank storage is established, and the influence law of the circumferential effect of the coil on the heating process of waxy crude oil is put forward. The external environment influence area, the internal temperature rise heat flow influence area and the transition area are put forward. The multi-nonlinear regression method is innovatively used to establish the main influencing factor model for each influence area so as to realize the quantitative characterization of the influence mechanism between the heat flux at the tank boundary and the dynamic thermal environment and the variable physical properties parameters of crude oil. The results show that the convective heat transfer of oil in the tank bottom area is greatly influenced by the circumferential effect of the coil, and the circumferential effect of the coil is enhanced with the gradual proximity of the coils on both sides, and the low temperature area at the tank bottom is obviously improved. The heat loss of the storage tank is mainly concentrated at the boundary of the tank. According to the change of heat flux in each boundary area, the external environment influence area, the internal heating heat flow influence area and the transition area are divided. Among them, the top of the storage tank forms an external environmental impact area dominated by solar radiation heat transfer, supplemented by natural convection of temperature difference, and the bottom of the storage tank forms an external soil impact area dominated by heat conduction to the external soil. Due to the effect of the thermal insulation layer, the influence of the external dynamic thermal environment on the position of the tank wall is weakened, and only the influence area of the internal temperature rise heat flow dominated by the natural convection of crude oil is formed. At the same time, there is a transition area with different thickness between the internal and external influence areas, which is affected by the external dynamic thermal boundary conditions and the physical properties of crude oil. The internal heat transfer direction is different, which causes the influence weight of each factor to fluctuate, and with the reduction of the external dynamic thermal environment, the transition area becomes the external condition influence area.

Displacement response characteristics of different sand tunnel excavation faces under true triaxial loading

During the construction of subway tunnels, the geotechnical body is affected by excavation to produce three-dimensional spatial deformation. The deformation of geotechnical bodies is an important safety hazard for project advancement, so it is important to understand the excavation disturbance range and deformation mechanism. The current related research is mainly about the theory of land subsidence and the two-dimensional plane displacement of the stratum, and there are few studies on the specific three-dimensional disturbance mode and its mechanism. In order to better understand the three-dimensional displacement characteristics of the tunnel excavation soil, a tunnel excavation model test was established based on a true triaxial stress loading system combined with three-dimensional scanning technology for a superimposed sandy soil. Based on the established model, the vector displacement response range and three-dimensional deformation characteristics of the excavation face were studied in the main displacement affected area around the excavation face. Meanwhile, the deformation characteristics, such as vertical settlement and horizontal displacement of the stratum in the main influence area were analyzed. The results show that the main influence area of tunnel excavation is elliptical and distributed within a range of twice the diameter of the tunnel axis. The main influence range is bell-shaped in the vertical direction and inverted wedge-shaped in the horizontal direction. The three-dimensional space presents a “W” deformation distribution. The three-dimensional deformation theoretical model of the excavation face established in this paper can provide some references for the excavation engineering of superimposed sand-soil tunnels.

Research on the Control Process of Mining and Combustion Disturbance Zone Based on the Three-Dimensional Dynamic Distribution Model of Void Fraction

ABSTRACT The cavities formed by the combustion of underground coal fires and the overlying rock damage caused by the combustion will cause the surface to subside or even collapse, forming crisscrossing surface cracks. On the basis of studying the fracture, subsidence and caving characteristics of overlying strata in the coal seam mining-combustion composite open space, the temporal and spatial development of the coal seam mining-combustion composite open space and the three-dimensional movement law of the induced overlying rock are studied in order to build three-dimensional dynamic theoretical distribution model of porosity in coal seam mining and combustion disturbance zone. Based on the porosity evolution model, the fire zone isotope radon method is used to detect the temperature anomalous area in the goaf. According to the location of the abnormal high-temperature area, a comprehensive treatment method is carried out, which is mainly based on the combined grouting of the upper and lower the mine, and supplemented by nitrogen injection and three-phase foam injection, thereby to solve the problem of difficult quantitative characterization of the multi-void medium in the mining-influenced underground coal fire area and targeted treatment for coal fire prevention and control in the mining-combustion disturbance area.

An analytical solution for the dynamic tunnel–soil–shallow foundation interaction under a harmonic point load

Elastic waves excited by railway tunnels can induce excessive vibration in nearby buildings. The dynamic interaction between the tunnel, soils, and the shallow foundations of buildings can significantly affect wave transmission from the vibration source to buildings. In this study, a three-dimensional theoretical model involving structure–soil–structure interaction (SSSI) is proposed and the analytical solution is obtained. The foundation is assumed to be seated on the ground surface, and the full coupling between the soil and structures is considered through the perfect contact at the tunnel-soil and soil-foundation interfaces. The response of the foundation to an incident wave field generated by a harmonic point load in a tunnel is calculated, and the impact of the tunnel–soil–foundation interaction is evaluated. The numerical results show that the SSSI needs to be considered when the relative distance between the tunnel and the foundation is within twice the shear wavelength. The vertical compliance coefficient at the center of the foundation features a significant peak due to the existence of a tunnel at the resonance frequency of the soil-tunnel system. The existence of the foundation restrains the displacement response of the ground surface within the foundation area, while causing significant amplification of the surface displacement outside the foundation area.

Do the photovoltaic poverty alleviation programs alleviate local energy poverty? —Empirical evidence of 9 counties in rural China

As an innovation of the Chinese government energy policy, solar photovoltaic poverty alleviation projects (PPAPs) play a positive role in solving energy shortages and poverty reduction. However, there is limited research on energy poverty in China's PPAP areas. This study has proposed a three-dimensional theoretical model for energy poverty, evaluated the energy poverty in the PPAP areas and analyzed its influencing path. The results show that: (1) The implementation of PPAPs has improved energy poverty to various degrees in all three types of solar resource areas; (2) The availability of energy services is the most prominent problem in energy poverty, and the PPAPs improve the availability of energy services the most; (3) Farmers' economic status and education have the greatest impact on alleviating energy poverty, and (4) The PPAPs directly improve the availability of energy services by increasing power generation and stability of the grid, and indirectly affect the cleanliness of energy consumption and energy affordability through farmers per se, thus affecting local energy poverty. Thus, public policies for promoting clean energy and energy utilization technologies (such as solar PV, etc) would help alleviate energy poverty in the rural areas.

A three-dimensional theoretical model of free vibration for multifunctional sandwich plates with honeycomb-corrugated hybrid cores

Combining honeycomb and corrugation to construct a multifunctional hybrid core for all-metallic sandwich construction can significantly enhance not only its stiffness and strength but also energy absorption and sound absorption relative to its honeycomb or corrugated sandwich counterpart of equal mass. However, how hybridization affects its vibration characteristics remains poorly understood, especially from a theoretical point of view. To address this deficiency, this study develops a theoretical model based on three-dimensional (3D) elasticity such that the free vibration of a hybrid-cored sandwich plate with arbitrary boundary conditions can be systematically evaluated. Based on micromechanics analysis of the representative volume element, the effect of honeycomb-corrugated coupling is duly accounted and the incorrectly expressed equivalent constitutive of hexagonal honeycombs in previous studies are modified. The model is validated against 3D finite element simulations. The natural frequencies of a honeycomb-corrugated sandwich construction are consistently higher than its honeycomb or corrugated sandwich counterparts, due mainly to mechanisms: honeycomb filling suppresses local vibration of face sheets and corrugated members, and mutual constraint of honeycomb and corrugation enhance the flexural rigidity of hybrid core. The proposed theoretical model can be easily extended to analyze other cases, such as sound radiation and sound insulation of hybrid sandwiches.

Three-Dimensional Stochastic Distribution Characteristics of Void Fraction in Longwall Mining-Disturbed Overburden of Inclined Coal Seam

Abstract Fractures in the overburden induced by mining disturbances provide a channel for fluid flow between the surface and the underground. Mining-induced strata movement and fracture distribution are influenced by the gravity and dip angles of rock seams. In this paper, a new three-dimensional theoretical distribution model for void fraction in each partition of overlying rock strata disturbed by inclined coal seam mining was constructed. Based on the theoretical determination model, the three-dimensional random distribution characteristics for void fraction were obtained by combining the random distribution law of void fraction obtained by similar physical simulation experiments and image processing techniques. Theoretical deterministic models, stochastic theoretical models, and similar physical simulations all show that void fraction distribution in the tendency direction of the coal seam shows a bimodal asymmetric distribution with high and low peaks and a symmetric distribution in the strike direction. The void fraction of the overburden in the central part of the mining area is smaller than that of the surrounding area. The results of the theoretically determined model and stochastic model of the void fraction for the strata with different mining lengths and different coal seam inclinations were compared with the results of similar simulation experiments, respectively. The results are in agreement, further verifying the practicality of the model.

Three-Dimensional Theoretical Model for Soil Arching with Inclined Slip Surfaces

Terzaghi proposed two-dimensional (2D) arching theory through trapdoor tests based on the assumption of a vertical slip surfaces. However, the original 2D assumption is different from a real three-dimensional (3D) excavation condition and actual slip surfaces caused by the downward movement of a trapdoor are inclined surfaces. Therefore, a 3D theoretical model was proposed in this study considering inclined slip surfaces. Horizontal thin-layer differential element method was used to obtain the loosening soil pressure. Using the 3D theoretical model, the effects of the buried depth ratio of loose area, length of loose area, soil parameters, and lateral earth pressure coefficient were investigated. The loosening earth pressure was highly affected by inclination angle of slip surface, buried depth ratio and length of loose area. Neglecting the inclined slip surface will underestimate the value of the loosening earth pressure, which will lead to the insecurity of the design. Loosening earth pressure calculated by this study was also compared with trapdoor tests. Results from this study was in good agreement with the experimental results. Compared with the traditional 2D solution, results from this study can more accurately analyze the soil arching effect in 3D excavation cases.

Vacuum Filling Simulation with Combined Lagranian and VOF Method

Jetting succeeded by accumulation is the characteristic of the vacuum filling, which is different from the conventional pressure-driven flow. In order to simulate this kind of flow, a three-dimensional theoretical model in terms of incompressible and viscous flow is established, and an iterative method combined with finite element method (FEM) is proposed to solve the flow problem. The Lagranian-VOF method is constructed to trace the jetting and accumulated flow fronts. Based on the proposed model and algorithm, a simulation program is developed to predict the velocity, pressure, temperature, and advancement progress. To validate the model and algorithm, a visual experimental equipment for vacuum filling is designed and construted. The vacuum filling experiments with different viscous materials and negative pressures were conducted and compared the corresponding simulations. The results show the flow front shape closely depends on the fluid viscosity and less relates to the vacuum pressure.

Three-dimensional theoretical analysis of seepage field in front of shield tunnel face

To evaluate hydraulic head distribution in front of a shield tunnel in a saturated soil layer, theoretical analysis and numerical simulations are carried out in this study. Based on the partial differential equilibrium equation of seepage flow, a three-dimensional (3D) theoretical analytical model of the shield tunnel face and the seepage field in front of it is established using the eigenfunction and the Fourier series expansion methods, and the hydraulic head calculation formula is derived. Combined with engineering cases, the theoretical analysis results and the 3D numerical simulation results are compared and analyzed. The effect of the water pressure of the tunnel face on the hydraulic head distribution is also analyzed. The results of the proposed analytical solution are in agreement with those of the numerical simulation solutions; moreover, the proposed analytical solution requires less time to calculate the seepage hydraulic head than the numerical simulation. The ratio of the initial water table to the diameter (D) of tunnel face has a more significant impact on the hydraulic head distribution at a position 0.5D above the tunnel vault. When the water pressure on the tunnel face is not considered, the values of the hydraulic head are significantly underestimated.