We present detailed simulations of black hole-neutron star (BH-NS) mergers kilonova and gamma-ray burst (GRB) afterglow and kilonova luminosity function, and discuss the detectability of electromagnetic (EM) counterpart in connection with gravitational wave (GW) detections, GW-triggered target-of-opportunity observations, and time-domain blind searches. The predicted absolute magnitude of the BH-NS kilonovae at $0.5,{rm days}$ after the merger falls in $[-10,-15.5]$. The simulated luminosity function contains the potential viewing-angle distribution information of the anisotropic kilonova emission. We simulate the GW detection rates, detectable distances and signal duration, for the future networks of 2nd/2.5th/3rd-generation GW detectors. BH-NSs tend to produce brighter kilonovae and afterglows if the BH has a higher aligned-spin, and a less massive NS with a stiffer EoS. The detectability of kilonova is especially sensitive to the BH spin. If BHs typically have low spins, the BH-NS EM counterparts are hard to discover. For the 2nd generation GW detector networks, a limiting magnitude of $m_{rm limit}sim23-24,{rm mag}$ is required to detect the kilonovae even if BH high spin is assumed. Thus, a plausible explanation for the lack of BH-NS associated kilonova detection during LIGO/Virgo O3 is that either there is no EM counterpart (plunging events), or the current follow-ups are too shallow. These observations still have the chance to detect the on-axis jet afterglow associated with an sGRB or an orphan afterglow. Follow-up observations can detect possible associated sGRB afterglows, from which kilonova signatures may be studied. For time-domain observations, a high-cadence search in redder filters is recommended to detect more BH-NS associated kilonovae and afterglows.
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