Potential Flow Region Research Articles

TURBULENCE MODELLING OF INCIPIENT WAVE BREAKING ON A VERTICAL CYLINDER ON A SLOPED BED

The extensive use of vertical piles as basic components of coastal structures, e.g. monopile wind turbine foundations, has made the study of wave impact on vertical cylindrical structures in the breaking zone of major practical importance. To investigate the breaking wave forces on a vertical cylinder located on a sloped bed, Reynolds-averaged Navier–Stokes equations (RANS) based turbulence models have been used as closure in most recent computational fluid dynamics (CFD) studies. The present study adopts a Reynolds stress model (RSM), namely the Wilcox (2006) stress-ww model, as our primary turbulence closure model to simulate the experiment of Irschik et al. (2004). The Wilcox (2006) stress-ww model has been analysed in the recent work of Li et al. (2022) and proven to be neutrally stable in the potential flow region beneath surface waves, thus it naturally avoids any unphysical turbulence over-production in the pre-breaking zone (unlike two-equation models in their standard forms). It has also shown good accuracy in predicting surf zone breaking waves in Li et al. (2022) and deep-water wave breaking due to modulational perturbations in Li and Fuhrman (2022). For comparison purposes, the present study also conducts simulations with the stabilized LF18 kk- ww model, as well as with no turbulence model.

Correction of GLS turbulence closure for wave-resolving models with stratification

We build on the work of Mayer and Madsen (2000) and Larsen and Fuhrman (2018) to correct for overmixing in wave potential flow regions due to the linear instability of two-equation closure systems. We extend it to the Generic Length Scale (GLS) turbulent closure approach, while testing it for the first time on a stratified nearshore problem. The modified turbulence equations affect only the nonbreaking wave potential flow region where the vorticity is less than 10% of the strain rate and thus can preserve stratification on the innershelf under favorable heat and wind forcing conditions. This work is a necessary step before engaging in full 3D wave-resolving studies of surf–shelf exchange.

Reynolds stress turbulence modelling of surf zone breaking waves

Computational fluid dynamics is increasingly used to investigate the inherently complicated phenomenon of wave breaking. To date, however, no single model has proved capable of accurately simulating the breaking process across the entirety of the surf zone for both spilling and plunging breakers. The present study newly considers the Reynolds stress–$\omega$turbulence closure model for this purpose, where$\omega$is the specific dissipation rate. Novel stability analysis proves that, unlike two-equation closures (at least in their standard forms), the stress–$\omega$model is neutrally stable in the idealized potential flow region beneath surface waves. It thus naturally avoids unphysical exponential growth of turbulence prior to breaking, which has plagued numerous prior studies. The analysis is confirmed through simulation of a progressive surface wave train. The stress–$\omega$model is then applied to simulate a turbulent wave boundary layer, demonstrating superior accuracy relative to a two-equation model, especially during flow deceleration. Finally, the stress–$\omega$model is employed to simulate spilling and plunging breaking waves, with seemingly unprecedented accuracy. Specifically, the present work marks the first time that a single turbulence closure model collectively: (1) avoids turbulence over-production prior to breaking, (2) accurately predicts the breaking point, (3) provides reasonable evolution of turbulent normal stresses, while also (4) yielding accurate evolution of undertow velocity structure and magnitude across the surf zone, for both spilling and plunging cases. Differences in the predicted Reynolds shear stresses (hence flow resistance) are identified as key to the improved inner surf zone performance, relative to a state-of-the-art two-equation model.

SIMULATING BREAKING WAVES WITH THE REYNOLDS STRESS TURBULENCE MODEL

The present study formally proved that Reynolds stress models (RSMs) are unconditionally stable in the potential flow regions. RSMs resolve all components of the Reynolds stress, eliminating e.g. the assumed isotropy of turbulence inherent within two-equation models. The present study implemented and applied Wilcox stress-omega turbulence model for simulating breaking waves. It shows that the Wilcox stress-omega turbulence model can predict accurate results from pre-breaking all the way into the inner surf zone, especially for the undertow velocity profiles in the inner surf zone, which even stabilized two-equation models fails to accurately predict.Recorded Presentation from the vICCE (YouTube Link): https://youtu.be/hZqqlGbHpkA

Bubbles and drops in the vicinity of turbulent/non-turbulent interface in turbulent boundary layers

The properties of low-vorticity pockets within the turbulent region (named as bubbles) and high-vorticity pockets within the potential flow region (named as drops) in the vicinity of turbulent/non-turbulent (T/NT) interface in turbulent boundary layers are first experimentally investigated using two-dimensional time-resolved particle image velocimetry. The T/NT interface is detected with a turbulent kinetic energy (TKE) criterion. The bubbles and drops which enlarge the T/NT interface area will promote the entrainment process. A detailed study revealed that the T/NT interface above bubbles is generally higher than the mean interface position, whereas the T/NT interface below drops is lower. Due to different sizes of bubbles and drops, though the temporal probability of bubbles is much larger than drops, the spatial probability is similar. Moreover, the conditional mean velocity distribution and the convection speed of the bubbles and drops are first obtained, especially in experiments. The streamwise velocity of fluids in bubbles at different Reynolds number is larger than surrounding turbulent flow, while in drops, the streamwise velocity is smaller than surrounding free-stream. The conditional mean streamwise velocity difference can be well scaled with respective friction velocity. Besides, bubbles and drops will convect downstream with the velocity of surrounding fluids.

Stabilized RANS simulation of surf zone kinematics and boundary layer processes beneath large-scale plunging waves over a breaker bar

This paper presents numerical simulations of a bichromatic wave group propagating and breaking over a fixed breaker bar. The simulations are performed using a newly stabilized Reynolds-averaged Navier–Stokes (RANS) two-equation turbulence closure, which solves the longstanding problem of over-production of turbulence beneath surface waves in the nearly potential flow region prior to breaking. This model has previously been tested on small-scale spilling breaking regular waves, whereas in this work focus is on full (rather than model) scale application, wave groups (rather than regular waves) and plunging (rather than spilling) breakers. Additionally this paper has novel emphasis on bottom boundary layer dynamics which are very important for cross-shore sediment transport predictions. The model is validated by comparing with results from a previous experimental campaign. The model is shown to predict the surface elevations, velocities and turbulence well in the shoaling and outer surf-zone, avoiding turbulence over-production and incorrect undertow structure typical of standard turbulence closures. Comparison with detailed boundary layer measurements in the shoaling position reveals that the model is able to accurately capture the temporal dynamics of the entire wave boundary layer, including evolution of the boundary layer thickness, velocity overshoot and phase-shifts. Comparison in the surf zone additionally reveals that the model is able to accurately capture the transport of breaking-induced turbulence into the wave boundary layer. The performance of the model indicates that it can be used directly in the simulation of cross-shore sediment transport and morphology and also be used to study important hydrodynamic processes, which can help improve the predictive skill of morphodynamic profile models applied in coastal engineering.

On the over-production of turbulence beneath surface waves in Reynolds-averaged Navier–Stokes models

In previous computational fluid dynamics studies of breaking waves, there has been a marked tendency to severely over-estimate turbulence levels, both pre- and post-breaking. This problem is most likely related to the previously described (though not sufficiently well recognized) conditional instability of widely used turbulence models when used to close Reynolds-averaged Navier–Stokes (RANS) equations in regions of nearly potential flow with finite strain, resulting in exponential growth of the turbulent kinetic energy and eddy viscosity. While this problem has been known for nearly 20 years, a suitable and fundamentally sound solution has yet to be developed. In this work it is demonstrated that virtually all commonly used two-equation turbulence closure models are unconditionally, rather than conditionally, unstable in such regions. A new formulation of the $k$–$\unicode[STIX]{x1D714}$ closure is developed which elegantly stabilizes the model in nearly potential flow regions, with modifications remaining passive in sheared flow regions, thus solving this long-standing problem. Computed results involving non-breaking waves demonstrate that the new stabilized closure enables nearly constant form wave propagation over long durations, avoiding the exponential growth of the eddy viscosity and inevitable wave decay exhibited by standard closures. Additional applications on breaking waves demonstrate that the new stabilized model avoids the unphysical generation of pre-breaking turbulence which widely plagues existing closures. The new model is demonstrated to be capable of predicting accurate pre- and post-breaking surface elevations, as well as turbulence and undertow velocity profiles, especially during transition from pre-breaking to the outer surf zone. Results in the inner surf zone are similar to standard closures. Similar methods for formally stabilizing other widely used closure models ($k$–$\unicode[STIX]{x1D714}$ and $k$–$\unicode[STIX]{x1D700}$ variants) are likewise developed, and it is recommended that these be utilized in future RANS simulations of surface waves. (In the above $k$ is the turbulent kinetic energy density, $\unicode[STIX]{x1D714}$ is the specific dissipation rate, and $\unicode[STIX]{x1D700}$ is the dissipation.)

Chaos in breaking waves

This study investigates the chaotic behavior of breaking waves by laboratory experiments and numerical modeling. Repeated laboratory runs with different initial velocity perturbations show that the wave profile before the wave breaks can be accurately reproduced, but the subsequent breaking process varies among runs, indicating the lack of repeatability of breaking waves in reality. Numerical simulations based on the Smoothed Particle Hydrodynamics method are further carried out to examine the repeatability of wave breaking process. Consistent with the laboratory observation, multiple numerical simulations with variations in initial conditions present highly repeatable velocity field and free surface profile in the potential flow region but considerable variation at the breaking and post-breaking processes. Comparison also shows that 3D vortex structures induced by breaking waves are different among cases. Analysis of particle trajectory reveals that there is a similar trajectory thus a minor trajectory divergence among particles that are initially located at the pre-breaking region and the flume bottom, which are not directly impacted by the breaking process. However, a much more significant particle trajectory divergence is observed among particles that are initially located at the wave-splash region and the bore propagation region. The rate of divergence of particle trajectory under breaking waves is further examined by computing the Lyapunov exponent, a widely used indicator of chaos. This study reveals that different initial velocity perturbations lead to variations of near-surface velocity at the onset of wave breaking, which eventually cause the development of drastically different breaking wave jets and splashes. Therefore, the process of wave breaking, like many other dynamic processes in nature, exhibits a chaotic behavior.

Identification and quantification of losses in a LPT cascade by POD applied to LES data

A POD based procedure has been developed to identify and account for the different contributions to the entropy production rate caused by the unsteady aerodynamics of a low-pressure (LP) turbine blade. LES data of the extensively studied T106A cascade have been used to clearly highlight the capability of POD to identify deterministic incoming wake related modes, stochastic fine-scale structures embedded within the bulk of the wake carried during migration, and coherent structures originating in the boundary layer as a consequence of the wake-boundary layer interaction process. The POD modes computed by a kinematic kernel generate a full and complete basis, where both the velocity and enthalpy fields have been projected through an extended POD procedure to determine the relative coefficients. This allows to separately compute orthogonal sets of contributions to turbulent kinetic energy production, enthalpy-velocity correlation and turbulent dissipation of resolved structures, thus clearly identifying the dominating modes (i.e. phenomena) responsible for the overall entropy production rate. Moreover, low-order truncation of these different contributions have been grouped into three different parts: those arising from the deterministic incoming wake, those due to the turbulence carried by the wakes and its interaction with the boundary layer, and those related to boundary layer events. The spatial integration of these low-order truncations restricted to the time-mean boundary layer, wake mixing and the potential flow regions of the blade passage allows gathering further information on the unsteady loss generation mechanisms, and where they mainly act. Particularly, results show that the procedure is able to decompose losses into the dominant contributions, thus providing a new tool for a rapid and clear identification of the different sources of losses in complex unsteady flow fields.

Accurate Estimation of Profile Losses and Analysis of Loss Generation Mechanisms in a Turbine Cascade

The paper analyzes losses and the loss generation mechanisms in a low-pressure turbine (LPT) cascade by proper orthogonal decomposition (POD) applied to measurements. Total pressure probes and time-resolved particle image velocimetry (TR-PIV) are used to determine the flow field and performance of the blade with steady and unsteady inflow conditions varying the flow incidence. The total pressure loss coefficient is computed by traversing two Kiel probes upstream and downstream of the cascade simultaneously. This procedure allows a very accurate estimation of the total pressure loss coefficient also in the potential flow region affected by incoming wake migration. The TR-PIV investigation concentrates on the aft portion of the suction side boundary layer downstream of peak suction. In this adverse pressure gradient region, the interaction between the wake and the boundary layer is the strongest, and it leads to the largest deviation from a steady loss mechanism. POD applied to this portion of the domain provides a statistical representation of the flow oscillations by splitting the effects induced by the different dynamics. The paper also describes how POD can dissect the loss generation mechanisms by separating the contributions to the Reynolds stress tensor from the different modes. The steady condition loss generation, driven by boundary layer streaks and separation, is augmented in the presence of incoming wakes by the wake–boundary layer interaction and by the wake dilation mechanism. Wake migration losses have been found to be almost insensitive to incidence variation between nominal and negative (up to −9 deg) while at positive incidence, the losses have a steep increase due to the alteration of the wake path induced by the different loading distribution.

Improvement and validation of a flow model for conical vortices

Separation bubble and conical vortices on a large-span flat roof were observed in this study through the use of flow visualization. The results indicated that separation bubble occurred when the flow was normal to the leading edge of the flat roof. Conical vortices that occur under the cornering flow were observed near the leading edge, and their appearance was influenced by the wind angle. When the wind changed from along the diagonal to deviating from the diagonal of the roof, the conical vortex close to the approaching flow changed from circular to be more oblong shaped. Based on the measured velocities in the conical vortices by flow visualization, a proposed two-dimensional vortex model was improved and validated by simplifying the velocity profile between the vortex and the potential flow region. Through measured velocities and parameters of vortices, the intensities of conical vortices and separation bubble on a large-span flat roof under different wind directions were provided. The quasi-steady theory was corrected by including the effect of vortices. With this improved two-dimensional vortex model and the corrected quasi-steady theory, the mean and peak suction beneath the cores of the conical vortices and separation bubble can be predicted, and these were verified by measured pressures on a larger-scale model of the flat roof.

Development and Verification of a Flow Model of Conical Vortices on Saddle Roofs

AbstractThrough particle image velocimetry, conical vortices were observed near the leading edge of a flat roof and two saddle roofs whose high points were windward. The influences of wind direction and roof curvature on the appearance of conical vortices are discussed in this paper. The results indicated that when the wind direction changed from along the diagonal to deviating from the diagonal of the roofs, the conical vortex close to the approaching flow changed, becoming more oblong. At the same wind direction, a greater roof curvature led to a larger angle between the vortex axis and the leading edge of the saddle roof. Based on velocities measured in conical vortices by flow visualization, a simplified two-dimensional vortex model was established by adding a transitional region between the vortex core and the potential flow region of the Rankine vortex. Through this flow model, the intensities of conical vortices on the two saddle roofs under different wind directions were deduced. The quasi-steady ...

Flow Similarity in Compressible Convex-Corner Flows

T HE Prandtl-Meyer expansion is well-known in supersonic convex-corner flows. It is also possible to formulate an interaction law in an explicit form that would relate the displacement effect of the boundary layer to the pressure induced in the inviscid part of subsonicflows. In the literature, studies that attributed theflow properties near the corner to the viscous-invisvid interaction at the transonic flow regime are scarce [1]. For a laminar boundary layer, an analysis by Ruban and Turkyilmaz [2] indicated that the displacement effect near the corner point is primarily due to the inviscid part. The flow is governed by the main part of the boundary layer and the potential flow region outside the boundary layer (inviscid-inviscid interaction). However, a subsequent study by Ruban et al. [3] highlighted the contribution of viscous-inviscid interaction, in which the displacement thickness near the corner is affected by the overlapping region that lies between the viscous sublayer and main part of the boundary layer. For a turbulent boundary layer, Chung [4] indicated that there would be a mild initial expansion, followed by a strong expansion near the corner apex and then downstream recompression for a compressible convex-corner flow. AtM 0:64 and 0.83, the flow is expanded from subsonic to transonic speed when the similarity parameter M 6:14. Note that the convex-corner angle ranges from 5 to 17 deg. A small separation bubble might also be born at the formation of ah normal shock wave [5]. With an increasing convexcorner angle atM 0:83, there is a slight upstreammovement of the separation position and a downstreammovement of the reattachment position. The shock-induced separated region at M 8:95 is expanded and also induces intense surface-pressure fluctuations, which are associated with the intermittent nature of the surface pressure signals or shock-excursion phenomena. The amplitude of peak surface-pressure fluctuations could also be scaledwithM [6]. In Chung’s study [4], it is known that M cannot be used as a similarity parameter for the test cases at M 0:34. However, it is also known that there exists a relationship (i.e., the Prandtl-Glauert rule) [7] between the minimum pressure coefficients for the incompressible airfoil Q2 Cp;ic and compressible airfoil Cp;c, where Cp;c Cp;ic= 1 M p . According to the hypothesis that small streamline deflections produce proportionally small changes in Mach number and pressure, a hodograph solution for compressible flow past a corner was given by Verhoff et al. [8]. The viscous sublayer is relatively unimportant when the Reynolds number is sufficiently large. In terms of the stream function , the hodograph equation for compressible corner flows is given as follows:

A New Single-blade Based Hybrid CFD Method for Hovering and Forward-flight Rotor Computation

A hybrid Euler/full potential/Lagrangian wake method, based on single-blade simulation, for predicting unsteady aerodynamic flow around helicopter rotors in hover and forward flight has been developed. In this method, an Euler solver is used to model the near wake evolution and transonic flow phenomena in the vicinity of the blade, and a full potential equation (FPE) is used to model the isentropic potential flow region far away from the rotor, while the wake effects of other blades and the far wake are incorporated into the flow solution as an induced inflow distribution using a Lagrangian based wake analysis. To further reduce the execution time, the computational fluid dynamics (CFD) solution and rotor wake analysis (including induced velocity update) are conducted parallelly, and a load balancing strategy is employed to account for the information exchange between two solvers. By the developed method, several hover and forward-flight cases on Caradonna-Tung and Helishape 7A rotors are performed. Good agreements of the loadings on blade surface with available measured data demonstrate the validation of the method. Also, the CPU time required for different computation runs is compared in the paper, and the results show that the present hybrid method is superior to conventional CFD method in time cost, and will be more efficient with the number of blades increasing.

Turbulent boundary layers and channels at moderate Reynolds numbers

The behaviour of the velocity and pressure fluctuations in the outer layers of wall-bounded turbulent flows is analysed by comparing a new simulation of the zero-pressure-gradient boundary layer with older simulations of channels. The 99 % boundary-layer thickness is used as a reasonable analogue of the channel half-width, but the two flows are found to be too different for the analogy to be complete. In agreement with previous results, it is found that the fluctuations of the transverse velocities and of the pressure are stronger in the boundary layer, and this is traced to the pressure fluctuations induced in the outer intermittent layer by the differences between the potential and rotational flow regions. The same effect is also shown to be responsible for the stronger wake component of the mean velocity profile in external flows, whose increased energy production is the ultimate reason for the stronger fluctuations. Contrary to some previous results by our group, and by others, the streamwise velocity fluctuations are also found to be higher in boundary layers, although the effect is weaker. Within the limitations of the non-parallel nature of the boundary layer, the wall-parallel scales of all the fluctuations are similar in both the flows, suggesting that the scale-selection mechanism resides just below the intermittent region, y/δ = 0.3–0.5. This is also the location of the largest differences in the intensities, although the limited Reynolds number of the boundary-layer simulation (Reθ ≈ 2000) prevents firm conclusions on the scaling of this location. The statistics of the new boundary layer are available from http://torroja.dmt.upm.es/ftp/blayers/.

Unsteady Mixed Convection Flow in the Stagnation Region of a Three Dimensional Body Embedded in a Porous Medium

An analysis is preformed to study the heat transfer characteristic of unsteady mixed convection flow of a viscous fluid in the vicinity of a stagnation point of a general three-dimensional body embedded in a porous media. The velocity in the potential flow is assumed to vary arbitrary with time. The non-Darcy effects including convective, boundary and inertial effects are included in the analysis. Both nodal-point region (0 ≤ c ≤ 1), where c = b/a is the ratio of the velocity gradients in y and x directions in the potential flow and saddle point region (−1 ≤ c < 0) are considered. The semisimilar solutions of the momentum and energy equations are obtained numerically using finite difference method. Also a self-similar solution is found when the velocity in the potential flow and the wall temperature vary with time in particular manner. Many results are obtained and a representative set is displayed graphically to illustrate the influence of the physical parameters on the surface shear stresses and the surface heat transfer.

Unsteady aerodynamic forces estimation on a square cylinder by TR-PIV

The unsteady aerodynamic forces acting on a square cross-sectional cylinder are investigated by means of time-resolved particle image velocimetry (TR-PIV) at Reynolds number 4,900. The objective of the investigation is to prove the feasibility of non-intrusive force measurements around two-dimensional bodies. The PIV measurements performed at a rate of 1 kHz enable a time resolved (TR) description of the vortex shedding phenomenon occurring at 10 Hz and to follow the time evolution of vortex dominated wake. The instantaneous aerodynamic force coefficients are obtained from the integration of the force equations within a control volume enclosing the object. The required instantaneous pressure distribution is inferred making use of two physical models: Bernoulli relation is adopted in the potential slowly-evolving flow region; in the turbulent wake, the Navier–Stokes equations are invoked to determine the pressure gradient spatial distribution, which integrated in space yields the pressure distribution. The spatial acceleration field is directly obtained from the temporal difference of the time-filtered velocity field. For a choice of the control volume approximately one model height away from the surface the contributions to the aerodynamic forces coming from the different terms of the force equation are individually examined. The convective term dominates the unsteady lift forces whereas the pressure term prevails for the drag. The temporal evolution of C L returns a clear periodic pattern in phase with the vortex shedding at a frequency of 10.1 Hz (Strouhal number St = 0.128) with oscillation amplitude of 0.9, whereas C D barely shows periodicity. The measurement uncertainties associated to the evaluation of all the terms in the force equation and especially in relation to TR-PIV measurements are discussed.

New Hybrid Method for Predicting the Flowfields of Helicopter Rotors

Based on a Navier-Stokes/full potential/free wake solver, a new hybrid method was developed for efficient predictions of the three-dimensional viscous flowfield of a helicopter rotor under both hover and forward flight conditions. The developed flow solver was composed of three modules: 1) a compressible Navier-Stokes analysis to model the viscous flow and near wake about the blade, 2) a compressible potential flow analysis to model the inviscid isentropic potential flow region far away from the rotor, and 3) a free wake model to account for tip vortex effects once the tip vortex leaves the viscous flow region and enters the potential flow region. In this hybrid method, a moving embedded grid methodology was adopted that accounts for rigid blade motions in rotation, flapping, and pitching. A dual-time method was employed to fulfill the calculation of the unsteady flowfields of helicopter rotors, and a third-order upwind scheme (MUSCL) and flux-difference splitting scheme without introducing artificial viscosity were used to calculate the flux. To search suitable donor elements in embedded grids to pass information between the viscous flow and potential flow zones, a new searching scheme was implemented. The sectional pressure distributions of a UH-60A helicopter rotor and an AH-1G model rotor in hover and forward flight, with and without the wake model, were calculated, and the developed hybrid model was validated by comparing with available experimental data. The simulated steady and unsteady lifting results of a model rotor with three different blade-tip planforms demonstrate the benefits of the curvilinear swept tip and constant swept tip in suppressing supercritical flows.

Damping of vertically excited surface wave in weakly viscous fluid

In a vertically oscillating circular cylindrical container, singular perturbation theory of two-time scale expansions is developed in weakly viscous fluids to investigate the motion of single free surface standing wave by linearizing the Navier-Stokes equation. The fluid field is divided into an outer potential flow region and an inner boundary layer region. The solutions of both two regions are obtained and a linear amplitude equation incorporating damping term and external excitation is derived. The condition to appear stable surface wave is obtained and the critical curve is determined. In addition, an analytical expression of damping coefficient is determined. Finally, the dispersion relation, which has been derived from the inviscid fluid approximation, is modified by adding linear damping. It is found that the modified results are reasonably closer to experimental results than former theory. Result shows that when forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when forcing frequency is high, the surface tension of the fluid is prominent.

Instability analysis of nonlinear surface waves in a circular cylindrical container subjected to a vertical excitation

Singular perturbation theory of two-time scale expansions was developed both in inviscid and weak viscous fluids to investigate the motion of single surface standing wave in a liquid-filled circular cylindrical vessel, which is subject to a vertical periodical oscillation. Firstly, it is assumed that the fluid in the circular cylindrical vessel is inviscid, incompressible and the motion is irrotational, a nonlinear evolution equation of slowly varying complex amplitude, which incorporates cubic nonlinear term, external excitation and the influence of surface tension, was derived from solvability condition of high-order approximation. It shows that when forced frequency is low, the effect of surface tension on mode selection of surface wave is not important. However, when forced frequency is high, the influence of surface tension is significant, and can not be neglected. This proved that the surface tension has the function, which causes free surface returning to equilibrium location. Theoretical results much close to experimental results when the surface tension is considered. In fact, the damping will appear in actual physical system due to dissipation of viscosity of fluid. Based upon weakly viscous fluids assumption, the fluid field was divided into an outer potential flow region and an inner boundary layer region. A linear amplitude equation of slowly varying complex amplitude, which incorporates damping term and external excitation, was derived from linearized Navier–Stokes equation. The analytical expression of damping coefficient was determined and the relation between damping and other related parameters (such as viscosity, forced amplitude and depth of fluid) was presented. The nonlinear amplitude equation and a dispersion, which had been derived from the inviscid fluid approximation, were modified by adding linear damping. It was found that the modified results much reasonably close to experimental results. Moreover, the influence both of the surface tension and the weak viscosity on the mode formation was described by comparing theoretical and experimental results. The results show that when the forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when the forcing frequency is high, the surface tension of the fluid is prominent. Finally, instability of the surface wave is analyzed and properties of the solutions of the modified amplitude equation are determined together with phase-plane trajectories. A necessary condition of forming stable surface wave is obtained and unstable regions are illustrated.